complex analysis ii - tu berlin · ii. 1.introduction the ﬁrst formal deﬁnition of a manifold...

Technische Universitat Berlin

Institut fur Mathematik

Complex Analysis II

Riemann surfaces

Prof. Dr. Ulrich Pinkall

Lecture notesDr. Felix Knoppel and Oliver Gross

February 12, 2019

Table of Contents

1 Introduction 11.1 Basic Concepts of Complex Analysis . . . . . . . . . . . . . . . 11.2 A Historical Interlude . . . . . . . . . . . . . . . . . . . . . . . 3

2 Complex Manifolds 82.1 Crash Course in Topology . . . . . . . . . . . . . . . . . . . . . 82.2 Smooth Structures . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Introduction to Several Complex Variables . . . . . . . . . . . 172.4 Complex Structures and Manifolds . . . . . . . . . . . . . . . . 222.5 Complex Linear Subspaces . . . . . . . . . . . . . . . . . . . . 24

3 The Tangent Bundle 273.1 Tangent Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 The tangent bundle as a smooth vector bundle . . . . . . . . . 323.3 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.4 The Lie-bracket of Vector Fields . . . . . . . . . . . . . . . . . . 38

4 Almost Complex Structures 404.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 The Newlander-Nirenberg Theorem . . . . . . . . . . . . . . . 424.3 Connections on Vector Bundles . . . . . . . . . . . . . . . . . . 434.4 Partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . 514.5 Conformal Equivalence . . . . . . . . . . . . . . . . . . . . . . 53

5 Integration on manifolds 565.1 Volume Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Integration of Forms . . . . . . . . . . . . . . . . . . . . . . . . 595.3 Stokes Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.4 Fundamental theorem for flat vector bundles . . . . . . . . . . 63

I

TABLE OF CONTENTS

6 Riemann surfaces 686.1 Holomorphic line bundles over a Riemann surface . . . . . . 716.2 Poincare-Hopf index theorem . . . . . . . . . . . . . . . . . . . 73

7 Classification of Line Bundles 857.1 Tensor products of vector spaces and bundles . . . . . . . . . 857.2 Line Bundles on Surfaces . . . . . . . . . . . . . . . . . . . . . 877.3 Combinatorial Topology . . . . . . . . . . . . . . . . . . . . . . 917.4 Discrete Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 967.5 Poincare Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 987.6 ”Baby Riemann Roch Theorem” . . . . . . . . . . . . . . . . . 1017.7 A Natural Complex Structure on Dual Spaces . . . . . . . . . 1057.8 Holomorphic Structures on Vector Bundles . . . . . . . . . . . 1067.9 Elliptic Differential Operators . . . . . . . . . . . . . . . . . . . 109

8 Appendix 122

II

1. Introduction

The first formal Definition of a manifold and a Riemann surface is found inthe paper ”Die Idee der Riemann’schen Flache” by H. Weyl in 1910.

1.1 Basic Concepts of Complex Analysis

The central objects of discussion in this course will be Riemann surfaces,but before we start we will revise some basic concepts and vivid ideas ofwhat we will be dealing with. Later we will then (hopefully) understandthat these, sometimes vaguely stated, claims are in fact true.

Within this course we will see that the easiest examples of Riemann surfacesare open sets U ⊂ C = R2, e.g.

D2 = (x, y) ∈ R2 | x2 + y2 < 1,

an annulus or the whole plane.

Definition 1.1 (homeomorphic). Let U, U ⊂ Rn are called homeomorphic,if there is a bijection ϕ : U → U such that ϕ and ϕ−1 both are continuous.

Example 1.2.

(i) The unit disc D2 and R2 are homeomorphic. A homeomorphism isgiven by

ϕ : (r cos φ, r sin φ) 7→ tan πr2 (cos φ, sin φ)

1

1.1 Basic Concepts of Complex Analysis

with inverse

ϕ−1 : (r cos φ, r sin φ) 7→ 2π arctan(r)(cos φ, sin φ).

It is easy to see that bot, ϕ and ϕ−1 are continuous.

(ii) U1 and U2 are homeomorphic.

(iii) U3 and U4 are homeomorphic.

Definition 1.3 (diffeomorphic). Let U, U ⊂ Rn open are called diffeomor-phic if there is a bijective map ϕ : U → U such that ϕ and ϕ−1 are bothsmooth.

The same statements as in example 1.2 about U1, . . . , U5, R2 are true replac-ing homeomorphic by diffeomorphic.

Definition 1.4 (biholomorphic). Let U, U ⊂ Rn open are called biholomor-phic if there is a bijective map ϕ : U → U such that ϕ and ϕ−1 are bothholomorphic.

Theorem 1.5. The open unit disc D2 is not biholomorphic to C.

Proof. Let ϕ : C → D2 be holomorphic. Then ϕ is bounded and hence byLiouville’s theorem constant. Thus ϕ is no bijection.

Theorem 1.6 (Riemann mapping theorem). Let U, U ⊂ C be open, U 6=C, U 6= C and both homeomorphic to D2. Then U is biholomorphic to U.

Remark 1.7. Considering theorem 1.6 one may get the idea that also theannuli U3 and U4 are biholomorphic. Surprisingly this does not hold ingeneral, but one can proof that indeed two annuli are biholomorphic if andonly if the quotient of their respective radii is equal.

Example 1.8 (Riemann sphere). An informal idea about Riemann surfacesis that they are ”surfaces” (2-dimensional manifolds) that ”locally” (meaningnear every point) look like an open set U ⊂ C. Another example of sucha surface is given by the Riemann sphere M = S2 = C. This can be seen byconsidering the stereographic projection from the north pole

σn : S2 \ (0, 0, 1)→ C

as indicated in figure 1.1.Using the projection from the south pole is changing the chart by z 7→ 1/z.Thus we have completely covered S2 by charts.

2

Introduction

Figure 1.1: An image of how the stereographic projection works.

1.2 A Historical Interlude

Riemann surfaces naturally came up in mathematics when trying to solveso called elliptic integrals. To get an idea about what an elliptic integral isand what makes them special, we will consider two examples that math-ematicians in early times tried to conquer. This will then lead to an ideahow elliptic integrals and Riemann surfaces fit together.

Ellipse: An ellipse is given by

E = (x, y) ∈ R2 | x2

a2 +y2

b2 = 1.

The length of E is then given by L = 4∫ a

0

»1 + f ′(x)2dx, where

f : [0, a]→ R, x 7→ f (x) = b…

1− x2

a2 .

3


Thenf ′(x) =

−xb

a2…

1− x2

a2

and thus, with z = x/a and e2 = 1− a2/b2,

L = 4a∫ 1

01−e2z2√

(1−z2)(1−e2z2)dz

which is an elliptic integral and has no solution by elementary func-tions.

Pendulum: Considering the (mathematical) pendulum two related prob-lems arise naturally:

(i) What is the period?

(ii) What is θ as a function of time?

We assume that m = g = L = 1. The pendulum equation is

θ′′ = − sin θ.

By adding − sin θ to both sides and multiplying the equation with θ′

we get θ′θ′′ + θ′ sin θ = 0.

We see that ÅÄ12 θ′ä2 − cos θ

ã′= θ′θ′′ + θ′ sin θ = 0.

ThusÄ

12 θ′ä2 − cos θ, the sum of the kinetic energy

Ä12 θ′ä2

and the po-tential energy 1− cos θ, must be constant.At maximum elongation α we have θ′ = 0, as the potential energy is

4

Introduction

maximal, what leads to const = − cos α. So we know the total energyof the system leading toÄ

12 θ′ä2

= cos θ − cos α ⇔ ddt θ = θ′ =

»2 (cos θ − cos α)

Thus, when θ′(t) ≥ 0, by seperation of variables

dt =dθ»

2(cos θ − cos α).

So that a quarter of a period, meaning from angle 0 to θ, is

t =∫ θ

0

dϕ√2(cos ϕ−cos α)

Withcos ϕ = cos2 ϕ

2 − sin2 ϕ2 = 1− sin2 ϕ

2 − sin2 ϕ2

we get1− cos ϕ = 2 sin2 ϕ

2 ,

thus as the same holds for cos α

t =∫ θ

0

dϕ√2(cos ϕ−cos α)

= 12

∫ θ

0

dϕ√sin2 α

2−sin2 ϕ2

= 12

∫ θ

01

sin α2

1Ã1−

sin2 ϕ2

sin2 α2

dϕ

Substitution with z =sin

ϕ2

sin α2

and defining e := sin α2 leads to

z =sin

ϕ2

e ; dϕ = 2ecos ϕ

2dz = 2e√

1−sin2 ϕ2

dz = 2e√1−e2z2 dz

what finally yields

t =∫ ρ

0dz√

(1−z2)(1−e2z2).

where ρ :=sin2 θ

2sin2 α

2. Thus, with α = θ it is ρ = 1, therefore the full

period is given by

T = 4∫ 1

0dz√

(1−z2)(1−e2z2).

5


But how does this now lead to Riemann surfaces? In two examples we havefound integrals of the form

∫ 1

0dz√

(1−z2)(1−e2z2),

∫ 1

0

(1−e2z2)√(1−z2)(1−e2z2)

dz

The map z 7→ (1− z2)(1− e2z2) has zeros at −1e ,−1, 1, 1

e . Define a := 1e .

What shall we do with some expression like√

z− a, say√

z? We have twochoices here.For each z ∈ C \ negative real axis consider two points representing ±

√z,

visualize a z, but in two ’sheets’ (copies of the complex plane).

The same procedure, this time we glue two copies of C together along thereal intervalls [−a,−1] and [1, a], turns

»(1− z2)(1− e2z2) into an honest

function on a Riemann surface – the two-sheeted cover of C branched over−a,−1, 1, a.

Remark 1.9. Note that the blue curve γ on the Riemann surface is closed,whereas the green η curve is not.

6

Introduction

Actually this can be regarded as a torus with two points removed – themissing points correspond to z = ∞.

7

2. Complex Manifolds

2.1 Crash Course in Topology

As a Riemann surface will be also a two-dimensional real manifold, whichitself is nothing more than a special kind of topological space it makes senseto shortly revise the most important definitions and ideas of Topology.

Definition 2.1 (Topological space). A topological space is a set M togetherwith O ⊂ P(M) (’collection of open sets’) such that

(i) ∅, M ∈ O

(ii) Uα ∈ O for α ∈ A⇒ ∪α∈AUα ∈ O

(iii) U1, . . . , Un ∈ O⇒ U1 ∩ · · · ∩Un ∈ O

Example 2.2 (subspace topology). If M ⊂ M , M topological space. ThenO = U ∩ M | U ∈ O defines a topology for M — the subset or relativetopology.

Example 2.3 (Quotient topology). If π : M → M, M topological space.Then O = U ∩ M | π−1U ∈ O defines a topology for M — the quo-tient topology. The map π yields an equivalence relation on M given byx ∼ y :⇔ π(x) = π(y). Conversely, the natural projection of ∼ is such amap π. We denote the quotient space by M = M/∼.

Example 2.4 (Torus). M = R2, p ∼ q :⇔ q− p ∈ Z2.

Definition 2.5 (Continuity). Let M, M be topological spaces. Then f : M→M is called continuous, if f−1(U) ∈ O for all U ∈ O.

8

Complex Manifolds

Definition 2.6 (Homeomorphism). Let M, M be topological spaces. Thenf : M → M is called a homoeomorphism, if f is bijective and f and f−1 bothare continuous.

To assure that the kind of topological spaces that we deal with have niceproperties and we do not have to deal with any horrific examples of topo-logical spaces that topologists can think of we have certain demands onthem.

Definition 2.7 (Hausdorff). A topological space M is called Hausdorff, if forall p, q ∈ M with p 6= q there are U, U ∈ O such that p ∈ U, q ∈ U andU ∩ U = ∅.

Theorem 2.8.

(i) Rn is Hausdorff

(ii) M Hausdorff, M ⊂ M. Then M is Hausdorff (with respect to the relativetopology).

Example 2.9. An example of a non-Hausdorff space is M = R togetherwith p ∼ q :⇔ q− p ∈ Q. Then M/∼ is not Hausdorff. You can check thatas an exercise.

In particular, quotient spaces tend not to be Hausdorff. So when we aredealing with quotient spaces it is always necessary to, at least, shortly thinkabout it.

Definition 2.10 (2nd-countable space). A topological space M is said tosatisfy the second axiom of countability (or, in short, is called 2nd countable)if there are U1, U2, U3, . . . ⊂ O such that for each U ∈ O there is I ⊂N suchthat U = ∪i∈IUi.

Theorem 2.11.

(i) Rn is 2nd-countable.

(ii) If M is 2nd-countable, M ⊂ M. Then M is 2nd-countable (with respectto the subspace topology).

Proof. (only for a)) Let U1, U2, . . . be an enumeration of all open balls in Rn

with center Qn and radius r ∈ Q. Then this is a basis of ORn .

9

2.2 Smooth Structures

Definition 2.12 (topological manifold). A topological space M is called ann-dimensional topological manifold, if M is a 2nd-countable Haussdorf spaceand for every p ∈ M there is an open set U 3 p, an open set V ⊂ Rn and ahomeomorphism ϕ : U → V.

Definition 2.13 (coordinate chart, atlas). A pair (U, ϕ) as in definition2.12 is called coordinate chart on a topological manifold M.An collection (Uα, ϕα)α∈I of corrdinate charts on a topological manifold M iscalled an atlas if M = ∪α∈IUα.


Definition 2.14 (coordinate change). Given two charts ϕ : U → Rn andψ : V → Rn of a topological manifold M, then the map

f : ϕ(U ∩V)→ ψ(U ∩V) with f = ψ (ϕ|U ∩V)−1

is a homeomorphism, called the coordinate change or transition map.

10

Complex Manifolds

Definition 2.15 (smoothly compatible). Two charts (U, ϕ), (V, ψ) on atopological manifold M are called smoothly compatible if the correspondingcoordinate change f = ψ (ϕ|U ∩V)

−1 is a diffeomorphism.

Example 2.16. Consider M = Sn ⊂ Rn+1, and define charts as follows:For i = 0, . . . , n,

U±i = x ∈ S2 | ±xi > 0with

ϕ±i , : U±i → B, ϕ±i (x0, . . . , xn) = (x0, . . . , “xi, . . . , xn),

where the hat means omission. To check that ϕi are homeomorphismswe make ourselves aware of the fact that this, as a projection, is indeedcontinuous and that inverse is given by the continuous mapÄ

ϕ±iä−1

(x0, . . . , “xi, . . . , xn) =(

x0, . . . ,√

1−∑j 6=i x2j , . . . , xn

).

Since Sn, as a subset of Rn+1, is Hausdorff and second countable, Sn is ann-dimensional topological manifold. All ϕ±i are compatible, so this atlaswith 2n + 2 charts turns Sn into a smooth manifold.

Figure 2.1: This illustration for the case n = 2 is taken from the title page ofthe book ”Riemannian Geometry” by Manfredo do Carmo (Birkenhauser1979).

As compatibility defines an equivalence relation on charts, the followingdefinition makes sense.

Definition 2.17 (maximal atlas). An atlas (Uα, ϕα)α∈I of mutually com-patible charts on M is called maximal if every chart (U, ϕ) on M which iscompatible with all charts in (Uα, ϕα)α∈I is already contained in the atlas.

Remark 2.18. A maximal atlas of mutually compatible charts is also oftencalled smooth structure on M.

11


Definition 2.19 (smooth manifold). A smooth structure defined on a topo-logical manifold M turns M into a smooth manifold.

Example 2.20.

1. Let M ⊂ Rn open with the chart (U, ϕ) where U = M and ϕ = idRn ,then this defines a smooth manifold.

2. (Product manifolds):Let M and N be topological manifolds of dimension m and n, respec-tively. Then their Cartesian product M×N together with the producttopology is a topological manifold of dimension m + n.Further, if (Uα, ϕα)α∈A is a smooth atlas of M and (Vβ, ψβ)β∈B is asmooth atlas of N, then (Uα ×Vβ, ϕα × ψβ)(α,β)∈A×B is a smooth atlasof M×N.Here

ϕα × ψβ : Uα ×Vβ → ϕα(Uα)× ψβ(Vβ)

is defined byϕα × ψβ(p, q) := (ϕα(p), ψβ(q)).

3. (Quotient spaces):Let M be a topological space and π : M→ M surjective. Define

O :=¶

U ⊂ M | π−1(U) ∈ O©

then this defines the quotient topology on M, i.e. M is also a topologicalspace now.Define an equivalence relation ∼ on M by

p ∼ q :⇔ π(p) = π(q),

then M is in bijective correspondence to the equivalence classes [π(p)]on M, thus

M = M∼.

As mentioned before, in particular for quotient spaces, the Hausdorff prop-erty and the property of being second countable have to be carefully checkedeach time as quotient spaces tend to lack of these.

Example 2.21. Examples of manifolds that are obtained by quotient spacesare the following:

1. (Real projective space):

Let n ∈ N and M := Rn+1 \ 0. The quotient space RPn = M∼with equivalence relation given by

x ∼ y :⇐⇒ x = λy, λ ∈ R

is called the n-dimensional real projective space.

12

Complex Manifolds

2. (Torus):Let M = Rn and define

p ∼ q :⇔ p− q ∈ Zn.

Then M∼ is the n-dimensional Torus Tn and is a smooth manifold.For n = 2 this manifold is actually diffeomorphic to the donut-shapedTorus embedded in R3 and S1 × S1.

Definition 2.22 (Smooth map). Let M and M be smooth manifolds. Thena map f : M → M is called smooth if for every chart (U, ϕ) of M and everychart (V, ψ) of M the map

ψ f ϕ−1 : ϕ( f−1(V)∩U)→ ψ(V), x 7→ ψ( f (ϕ−1(x)))

is smooth.

Definition 2.23 (Diffeomorphism). Let M and M be smooth manifolds.Then a bijective map f : M → M is called a diffeomorphism if both f andf−1 are smooth.

Note that the property of being diffeomorphic defines an equivalence real-tion on smooth manifolds. Diffeomorphic manifolds are indistinguishablefrom the viewpoint of differential topology. In fact, the following interest-ing theorems hold.

13


Theorem 2.24. Every connected 1-dimensional smooth manifold is diffeomor-phic to either (0, 1), or S1.

Theorem 2.25. Every connected and compact 2-dimensional smooth manifoldis diffeomorphic to exactly one in the following list:

The poorly drawn manifolds in the second row of the theorem above aresupposed to be Boy’s surface, then the same with a hole and so on. But thisis quite hard to visualize. If we demand the manifolds to be orientable,then the theorem also holds, but only the first row of cases can appear.

Manifolds of dimension 3 or higher can sadly not be ordered by a list likein the lower dimensional cases. There are too many of them.

If we have a manifold of dimension k, it makes sense to think about whetherthe locally structure transfers to lower dimensional subsets. How this canbe the case is formalized in the following definition.

Definition 2.26 (Submanifold). A subset M ⊂ M in a k-dimensional smoothmanifold M is called an n-dimensional submanifold if for every point p ∈ Mthere is a chart ϕ : U → V of M with p ∈ U such that

ϕ(U ∩M) = V ∩(Rn × 0) ⊂ Rk.

14

Complex Manifolds

If we restrict ourselves to considering manifolds M = Rk, we have thefollowing equivalent definitions of a submanifold.

Theorem 2.27. Let M ⊂ Rk be a subset. Then the following are equivalent:

a) M is an n-dimensional submanifold,

b) locally M looks like the graph of a map from Rn to Rk−n, which means:For every point p ∈ M there are open sets V ⊂ Rn and W ⊂ M, W 3 p,a smooth map f : V → Rk−n and a coordinate permutation π : Rk → Rk,π(x1, ..., xk) = (xσ1 , ..., xσk) such that

π(W) = (x, f (x)) | x ∈ V,

c) locally M is the zero set of some smooth map into Rk−n, which means: Forevery p ∈ M there is an open set U ⊂ Rk, U 3 p and a smooth mapg : U → Rk−n such that

M∩U = x ∈ U | g(x) = 0

and the Jacobian g′(x) has full rank for all x ∈ M,

d) locally M can be parametrized by open sets in Rn, which means: For everyp ∈ M there are open sets W ⊂ M, W 3 p, V ⊂ Rn and a smooth mapψ : V → Rk such that ψ maps V bijectively onto W and ψ′(x) has fullrank for all x ∈ V.

Proof.

(b)⇒ (a): Let p ∈ M. By b) after reordering coordinates in Rk we findopen sets V ∈ Rn, W ⊂ Rk−n such that p ∈ V ×W and we find a

15


smooth map f : V → W such that V ×W ∩M = (x, f (x)) | x ∈ V.Then

ϕ : V ×W → Rk, (x, y) 7→ (x, y− f (x))

is a diffeomorphism and ϕ(M∩(V ×W)) ⊂ Rn × 0.

(a)⇒ (c): Let p ∈ M. By a) we find an open U ∈ Rk, U 3 p and a diffeo-morphism ϕ : U → U ⊂ Rk such that ϕ(U ∩M) ⊂ Rn × 0. Nowdefine g : U → Rk−n to be the last k − n component functions of ϕ,i.e. ϕ = (ϕ1, . . . , ϕn, g1, . . . , gk−n). Then M∩(V ×W) = g−1(0). Forq ∈ V ×W we have

ϕ′(q) =

∗...∗

g′1(q)...

g′k−n(q)

.

Hence g′ has rank k− n.

(c)⇒ (b): This is just the implicit function theorem.

(b)⇒ (d): Let p ∈ M. After reordering the coordinates by b) we have anopen neighborhood of p of the form V×W and a smooth map f : V →W such that

M∩(V ×W) = (x, f (x)) | x ∈ V.

Now define ψ : V → Rk by ψ(x) = (x, f (x)), then ψ is smooth

ψ′(x) =Ç

IdRn

f ′(x)

åSo ψ′(x) has rank n for all x ∈ V. Moreover, ψ(V) = M× (V ×W).

(d)⇒ (b): Let p ∈ M. Then by d) there are open sets V ⊂ Rn, U ⊂ Rk, U 3p and a smooth map ψ : V → Rk such that ψ(V) = M∩U such thatrank ψ′(x) is n for all x ∈ V. After reordering the coordinates on Rk

we can assume that ψ = (φ, f )t with φ : V → Rn with det φ′(x0) 6= 0,where ψ(x0) = p. Passing to a smaller neighborhood V ⊂ V, V 3 p,we then achieve that φ : V → φ(V) is a diffeomorphism (by the inversefunction theorem). Now for all y ∈ φ(V) we have

ψ(φ−1(y)) =(

φ(φ−1(y)f (φ−1(y))

)=:

(y

f (φ−1(y))

)

16

Complex Manifolds

Now we will address ourselves to the topic of Riemann surfaces. To do thatwe first need to translate the just defined machinery to holomorphic maps.Most of the definitions can be adopted word by word only changing smoothor diffeomorphic to holomorphic or biholomorphic. But before we do this wewill first need to define holomorphicity for higher dimensions.

2.3 Introduction to Several Complex Variables

Every n-dimensional complex vector space V is automatically a 2n-dimensionalreal vector space:If v1, . . . , vn is a complex basis of V, then v1, iv1, . . . , vn, ivn is a real basis ofV. On V as a real vector space we have a real linear map

J : V → V, v 7→ Jv = iv

(complex scalar multiplication) and J satisfies J2 = −Id.

Conversely: If V is a real vector space with an R-linear map J : V → Vsuch that J2 = −Id, then we can make V into a complex vector space bydefining a complex scalar multiplication by

(α + iβ)v := (α + βJ)v .

Corollary 2.28. No such J exists on a real vector space of odd dimension.

Definition 2.29 (multi-variable holomorphicity). A map f : U → Ck, forU ⊂ Cn open, is called holomorphic if is continuously differentiable in the realsense and for all p ∈ U the linear map f ′(p) : R2n → R2k is complex-linear.

Remark 2.30. An R-linear map is also complex-linear if

A(JR2n v) = JR2k(Av),

i.e. if AJ = JA holds.

Theorem 2.31. Let (V, JV) and (W, JW) be two complex vector spaces. Everyreal linear map A : V → W can be uniquely decomposed into A = A′ + A′′ .where A′ is a complex-linear map

A′ : V →W such that A′ JV = JW A′

and A′′ is a complex-anti-linear map

A′′ : V →W such that A′′ JV = −JW A′′

17


Proof. Define A′ = 12(A− JW AJV) and A′ = 1

2(A + JW AJV) and check A′ iscomplex-linear and A′′ is complex-anti-linear. Uniqueness is clear.

Theorem 2.32. Let U ⊂ Cn and V ⊂ Ck be open, f : U → Ck, h : V → U.Then:

(i) f = ( f1, . . . , fk) is holomorphic if and only if f j : U → C are holomorphicfor all j = 1, . . . , k.

(ii) If f , g : U → C holomorphic then f + g and f · g are holomorphic and ifg(z) 6= 0 for all z ∈ U then f /g is holomorphic.

(iii) g, h holomorphic then g h is holomorphic.

Proof. The proof of this theorem will be a homework exercise.

Multiindices: Dealing with functions in several complex variables can of-ten lead to a mess when it comes to indexing. A neat way of dealingwith this problem is to introduce so called multiindices that are de-fined as follows:Let z = (z1, . . . , zn) ∈ Cn, k = (k1, . . . , kn) ∈Nn, then

zk :=n∏

j=1z

kjj = zk1

1 · . . . · zknn .

If all zj 6= 0, then this works also for k ∈ Zn.

Let further be k ∈ Cn. Later terms of the form 1w−z

−1will appear.

These will have to be understood as

1w−z = (w− z)−1 = (w− z)(−1,...,−1) =

n∏j=1

1(wj−zj)

.

In a similar manner we will deal with expressions of the form k + 1,we simply add 1 to each of the components of k, i.e.

k + 1 = (k1 + 1, . . . , kn + 1) .

As we translate theorems of complex analysis of one variable to thehigher-dimensional case, we will also have to use integration, thusexpressions of the form dw will appear. For w ∈ Cn these are definedas

dw := dw1 . . . dwn.

18

Complex Manifolds

Definition 2.33 (Polydisc, n-Torus). For a ∈ Cn, r ∈ Rn, we define then-dimensional polydisc with center a and radius r by

Da,r := (z1, . . . , zn) ∈ Cn | |zj − aj| < rj∀j = Da1,r1 × · · · × Dan,rn

and the n-dimensional Torus with center a and radius r by

Tn(a, r) := (z1, . . . , zn) ∈ Cn | |zj − aj| = rj∀j = S1a1,r1× · · · × S1

an,rn

Remark 2.34. Note that Tn is not the actual topological boundary of Da,r inCn. Anyways, it will act like it in the sense that it will be the right choiceto generalize one-dimensional theorems into higher dimensions.

Theorem 2.35 (Cauchy formula). Let U ⊂ Cn be open, Da,r ⊂ U andf : U → C holomorphic. Then for all z ∈ Da,r

f (z) = 1(2πi)n

∫Tn(a,r)

f (w)w−z dw.

Proof. Proof by induction on n ∈N:For n = 1 that’s just the well-known Cauchy formula.For the induction step we just consider the function g = f (·, z2, . . . , zn)defined on Da1,r1 . Then, by Cauchy formula applied to g and inductionhypothesis and by Fubini,

f (z1, . . . , zn) =1

2πi

∫S1

a1,r1

f (w1,z1,...,zn)w1−z1

dw1

= 12πi

∫S1

a1,r1

1w1−z1

1(2πi)n−1

∫f (w1,...,wn)

(w2−z2)···(wn−zn)dw2 · · ·dwndw1

= 1(2πi)n

∫Tn(a,r)

f (w)w−z dw.

Theorem 2.36. Holomorphic maps f : → Ck, for U ⊂ Cn open, are smooth,i.e. all partial derivatives exist.

Proof. It is enough to consider only the case k = 1. Since by the Cauchyformula

f (z) = 1(2πi)n

∫Tn(a,r)

f (w)w−z dw,

we have∂ f∂zj

(z) = 1(2πi)n

∫Tn(a,r)

∂∂zj

f (w)w−z dw = 1

(2πi)n

∫Tn(a,r)

f (w)(wj−zj)(w−z)dw .

Thus, by induction, all partial derivatives exist. Holomorphicity followsfrom the holomorphicity of the integrand.

19


Some more multiindices: Let k = (k1, . . . , kn). Then, with

k! = k1! · · · kn!,

the following can be defined:

f (k)(z) := ∂k1+...+kn

∂z1k1 ···∂zn

kn = ∂k1

∂z1k1

. . . ∂kn

∂znkn f (z) = k!

(2πi)n

∫Tn(a,r)

f (w)(w−z)k+1 dw

Further, by k ≥ 0 we denote k ∈Nn.

Theorem 2.37 (Taylor/Power series expansion). The Taylor series of a holo-morphic function f : Da,r → C converges everywhere to f :

f (z) =∑

k∈Nn

1k! f (k)(a)(z− a)k.

Remark 2.38. We will see that also the proof of the power series expansiontheorem works just like in the one-dimensional case again. But we need thefollowing observation:It is known that the formula for the geometric series holds in C, i.e. for|q| < 1 it is

∞∑k=0

qk =1

1− q.

Using this, for |z| < |w| we get

1w− z

=1w

11− z

w=

∞∑k=0

zk

wk+1 .

For the multi-variable version we will make use of the absolute convergenceof the geometric series that will allow us to change the order of summation.For z, w ∈ Cn such that

∣∣∣zj∣∣∣ < ∣∣∣wj

∣∣∣ for all j = 1, . . . , n it is

1w− z

=1

w1 − z1. . .

1wn − zn

=∑k≥0

zk11

wk1+11

. . . zknn

wkn+1n

=∑k≥0

zk

wk+1

Proof. We will start with the right-hand side and plug in the definition off (k)(z), this yields:∑

k≥0

1k! f (k)(a)(z− a)k = 1

(2πi)n

∑k≥0

(z− a)k∫

Tna,r

f (w)(w−a)k+1 dw

= 1(2πi)n

∫Tn

a,rf (w)

Ñ∑k≥0

(z−a)k

(w−a)k+1

édw

= 1(2πi)n

∫Tn

a,r

f (w)(w−a)−(z−a)dw

= 1(2πi)n

∫Tn

a,r

f (w)(w−z)dw

= f (z)

20

Complex Manifolds

where we used the observation of remark 2.38 and for the last equality theCauchy integral theorem.

Corollary 2.39 (Principle of analytic continuation). Let U ⊂ Cn open andconnected, ∅ 6= V ⊂ U open and f : U → C holomorphic with f |V = 0. Thenf (z) = 0 for all z ∈ U.

Proof. DefineE :=

⋂k∈Zn

z ∈ U | f (k)(z) = 0

then E is a closed subset of U and V ⊂ E. Further, by the power seriesexpansion theorem, E is open. Since U is connected we can deduce thatU = ∅ or E = U. Since E 6= ∅ by assumption, it it E = U. Thus the powerseries theorem implies that f ≡ 0.

Remark 2.40. In the one dimensional case we know that it even suffices toknow that f (z) = 0 for on set V ⊂ U that has an accumulation point in U.This does not generalize to higher dimensions. An easy counter-examplefor this is f (z1, z2) = z1. Then f 6= 0 but E = z1 = 0 certainly has anaccumulation point.

The power series expansion theorem and the corollary do not hold in a realC∞-setting. There the situation is utterly different. There we can work witha toolbox of functions that are all C∞ but are constant on certain domains.

Toolbox of C∞-functions:Consider f : R→ R with

f (x) =

0 for x ≤ 0,e−1/x for x > 0.

This certainly is C ∞ and so is then

g(x) = f (1− x2)

andh(x) =

∫ x

0g.

From h this we can build a smooth function h : R→ [0, 1] with h(x) =1 for x ∈ [−1

4 , 14 ] and h(x) = 0 for x ∈ R \ (−1, 1).

21

2.4 Complex Structures and Manifolds

There are also higher dimensional versions: We can define a smoothfunction

h : Rn → R, h(x) = h(x21 + · · ·+ x2

n)

which vanishes outside the unit ball and is constant ≡ 1 inside theball of radius 1

2 .

2.4 Complex Structures and Manifolds

Now we are able to define what complex manifold is supposed to be. Asalready mentioned, most of the definitions translate pretty much directlyfrom the smooth setting.

Definition 2.41 (holomorphically compatible). Two charts (U, ϕ), (V, ψ)on a 2n-dimensional topological manifold M are called holomorphically com-patible if the corresponding coordinate change f = ψ (ϕ|U ∩V)

−1 is biholo-morphic.

Definition 2.42 (complex structure). A complex structure on a topologicalmanifold M is a maximal atlas of holomorphically compatible charts.

Definition 2.43 (Complex manifold). An n-dimensional complex manifoldM is a 2n-dimensional topological manifold together with a complex structure.

Example 2.44 (Riemann sphere).

22

Complex Manifolds

Conisder the Riemann sphere C = C ∪ ∞ ' S1 together with the stere-ographic projection σN from the North Pole and σS from the South Pole.Then, (σN, σS) form an atlas of S1 and the transition map σS σ−1

N = 1z is

holomoprhic, hence the equivalence class of mutually compatible maps onC defines a complex structure on the Riemann sphere. This turns C into acomplex manifold.

Definition 2.45 (Holomorphic map). Let M and M be complex manifolds.Then a map f : M → M is called smooth if for every chart (U, ϕ) of M andevery chart (V, ψ) of M the map

ψ f ϕ−1 : ϕ( f−1(V)∩U)→ ψ(V), x 7→ ψ( f (ϕ−1(x)))

is holomorphic.

Definition 2.46 (biholomorphic). Two complex manifolds M, M are calledbiholomorphic if there is a biholomorphism between them.

Example 2.47.

(i) Any open subset M ⊂ Cn is a complex manifold.

(ii) (Complex Torus)Let a1, . . . , a2n be a real basis of R2n ∼= Cn. Define

Γ := z1a1 + . . . , z2na2n | z1, . . . , z2n ∈ Z ,

23

2.5 Complex Linear Subspaces

then Γ is called an integer-Lattice in R2n. Then

M := CnΓ := Cn

∼with z ∼ w :⇔ z − w ∈ Γ together with the quotient topology is atopological manifold. The obvious charts are for U ⊂ Cn such thatz− w /∈ Γ for all z, w ∈ U given by

ϕ : U → Cn, ϕ(z) = z.

Then coordinate changes are of the form z 7→ z + γ for γ ∈ Γ, thusin particular holomorphic. Taking the maximal atlas turns M into ann-dimensional complex manifold.

Complex analysis deals with classifying complex manifolds up to biholo-morphy and studying their properties. In this class we will mainly dealwith one complex-dimensional manifolds, thus they get a special name.

Definition 2.48 (Riemann surface). A complex manifold with dimC M = 1is called a Riemann surface.

As a motivation why one should study Riemann surfaces we already con-sidered elliptic integrals. Doing so, the problem of how to deal with ex-pressions of the form w =

»(1− z2)(1− e2z2) arouse. If we consider the

complex curve

M =¶(z, w) ∈ C2 | w2 = (1− z2)(1− e2z2)

©then this will be a Riemann surface. This is due to the implicit functiontheorem, but to get a feeling why this is true, have a look in the Appendixon the topic of algebraic curves.


Knowing what a complex manifold is immediately raises the question ofhow the notion of a submanifold is possible in the complex setting. It turnsout, that it works in a similar manner, so that we even find a complex equiv-alent of theorem 2.27.

In the turtorials (cf. Appendix) we have already learned about linear com-plex structures, so we will shortly recall the following.

24

Complex Manifolds

Definition 2.49 (Complex vector space). A complex n-dimensional vectorspace is a real 2n-dimensional vector space with J ∈ End(V) with J2 = −Id.

Definition 2.50 (Complex subspace). A linear subspace (in the real sense)U of a complex vector space V is called complex subspace if JU ⊂ U, i.e. it isJ-invariant.

We will introduce the following notation that will come in handy for ourfurther purposes.

HomR(V, W) = g : V →W | g linear in the real sense

Definition 2.51 (Complex linear vector space). Let V, W be complex vectorspaces. Then f ∈ HomR(V, W) is called complex linear if JW f = f JV .

Denote the set of complex linear homomorphisms as

HomC(V, W) = f ∈ Hom(V, W) | JW f = f JV.

If V, W are complex vector spaces, then the complex structure JV⊕RW onV ⊕R W, where ⊕R denotes the real direct sum, is given by

JV⊕RW(v, w) = (JVv, JWw).

Moreover, the complex structure JHomC(V,W) of HomC(V, W) is given by

JHomC(V,W) f = JW f = f JV .

Theorem 2.52. Let V, W be complex vector spaces. Then f ∈ HomR(V, W) iscomplex linear if and only if the graph G f = (v, f (v)) ∈ V ⊕R W | v ∈ Vof f is a complex subspace.

Proof.

”⇒”: For f ∈ HomC(V, W) we have

J(v, f (v)) = (Jv, J f (v)) = (Jv, f (Jv)) ∈ G f .

”⇐”: If G f is a complex subspace, then (Jv, J f (v)) = J(v, f (v)) ∈ G f , thusJ f (v) = f (Jv).

25


Remark 2.53. Let M ⊂ Rk be a submanifold.

Then M is locally the graph of a function f , p = (x, f (x)). Then we havethe ”pre-tangent space of M at p” given by

TpM := G f ′(p) = v, f ′(p)v | v ∈ Rn.

On the other hand M is locally the zero set of a function g. Then

TpM = kerg′(p).

Moreover, M is locally the image of a map ψ, ψ(x) = p. Then

TpM = Image(ψ′(p)).

Definition 2.54 (Complex submanifold). A 2n-dimensional submanifold ofR2k = Ck is called a complex submanifold if for each p ∈ M the pre-tangentspace TpM is a complex subspace of Ck.

Theorem 2.55. Let M ⊂ Ck be a subset. Then the following are equivalent:

a) M is an n-dimensional complex submanifold.

b) Locally M is the graph of a holomorphic function from an open subsetU ⊂ Cn into Ck−n.

c) Locally M is the zero set of a holomorphic g into Ck−n of full rank.

d) Locally M is the image of a holomorphic ψ from some open subset U ⊂ Cn

with full rank.

For all these equivalences, even in the real case, we rely on f ′ to exist. Forthe Ck setting this is fine, but in the general case, we are missing an ambientspace that allows us to think of tangent vectors. Remedy will be providedby the definition of a tangent space for the general setting.

26

3. The Tangent Bundle

3.1 Tangent Spaces

The space C∞(M) is a real vector space. Define tangent vectors X at p bytheir directional derivative operators.

Definition 3.1 (Tangent vector). A linear map X : C∞(M) → R is calleda tangent vector of M at p if there is a smooth curve γ : (−ε, ε) → M withγ(0) = p and X f = d

dt

∣∣∣t=0

f (γ(t)) = ( f γ)′(0).

The tangent space of M at p is the set of tangent vectors of M at p and willbe denoted by TpM, i.e.

TpM := X ∈ C∞(M)∗ | X is tangent vector of M at p

For a chart (U, ϕ), U 3 p, ϕ = (x1, . . . , xn) we define ∂∂xj

∣∣∣∣p∈ TpM byÇ

∂∂xj

∣∣∣∣p

åf := d

dt

∣∣∣t=0

f ϕ−1︸︷︷︸=: f

(q + tej) =: ∂ej f (q).

27

3.1 Tangent Spaces

Let X ∈ TpM defined by γ, γ = ϕ γ, γ′(0) = (a1, . . . , an). Then

X f = ( f γ)′(0) = ( f γ)′(0) = f ′(q)γ′(0) =∑

jaj

∂∂xj

∣∣∣∣p

f .

This shows that TpM is a linear subspace and

TpM = span ∂∂x1

∣∣∣p

, . . . ∂∂xn

∣∣∣p .

That the vectors ∂∂xj

∣∣∣∣p

are also linearly dependent follows from the follow-

ing lemma.

Lemma 3.2. For each a ∈ Rn there is a function f such that ∂∂xj

∣∣∣∣p

f = aj for

j = 1, . . . , n.

Proof. The existence of a locally defined f is clear — we can just use achart. Now we need to show that we can extend f to a function defined onM globally. This can be done using a so called bumb function, i.e. a non-negative smooth function ρ : M → [0, 1] which is constantly 1 on a smallneighborhood of p and zero outside the chart domain.

Theorem 3.3 (Transformation of coordinate frames). If (U, ϕ) and (V, ψ)are charts with p ∈ U ∩V, ϕ|U ∩V = Φ ψ|U ∩V . Then for every X ∈ TpM,

X =∑

ai∂

∂xi

∣∣∣∣∣p=∑

bi∂

∂yi

∣∣∣∣∣p

,

where ϕ = (x1, . . . , xn), ψ = (y1, . . . , yn), we haveÜa1...

an

ê= Φ′(ψ(p))

Üb1...

bn

ê.

28

The Tangent Bundle

Proof. Let γ : (−ε, ε) → M such that X f = ( f γ)′(0). Let γ = ϕ γ andγ = ψ γ, then

a = γ′(0), b = γ′(0).

Let Φ : ψ(U ∩V)→ ϕ(U ∩V) be the coordinate change Φ = ϕ ψ−1. Then

γ = ϕ γ = Φ ψ γ = Φ γ.

In particular,

a = γ′(0) = (Φ γ)′(0) = Φ′(ψ(p))γ′(0) = Φ′(ψ(p))b.

Definition 3.4 (Derivative as map TpM → Tf (p)M). Let M and M besmooth manifolds, f : M→ M smooth, p ∈ M. Then define a linear map

dp f : TpM→ T f (p)M

by setting for g ∈ C ∞(M) and X ∈ TpM

dp f (X)g := X(g f ).

Remark 3.5. dp f (X) is really a tangent vector in TpM because, if X corre-sponds to a curve γ : (−ε, ε)→ M with γ(0) = p then

dp f (X)g =ddt

∣∣∣∣∣t=0

(g f ) γ =ddt

∣∣∣∣∣t=0

g ( f γ︸︷︷︸=:γ

) =ddt

∣∣∣∣∣t=0

g γ.

Notation:The tangent vector X ∈ TpM corresponding to a curve γ : (−ε, ε)→ Mwith γ(0) = p is denoted by X =: γ′(0).

Theorem 3.6 (Chain rule). Suppose g : M → M, f : M → M are smoothmaps, then

dp( f g) = dg(p) f dpg.

29

3.1 Tangent Spaces

Definition 3.7 (Tangent bundle). The set

TM :=⊔

p∈MTpM

is called the tangent bundle of M. The map π : TM → M, TpM 3 X → p iscalled the projection map. So TpM = π−1(p).

Most elegant version of the chain rule:If f : M→ M is smooth, then d f : TM→ TM where

d f (X) = dπ(X) f (X).

So every X ∈ TM knows ”where it comes from”. With this notation,

d( f g) = d f dg.

Theorem 3.8. If f : M→ M is a diffeomorphism then for each p ∈ M the map

dp f : TpM→ T f (p)M

is a vector space isomorphism.

Proof. f is bijective and f−1 is smooth, IdM = f−1 f . For all p ∈ M,

IdTpM = dp(IdM) = d f (p) f−1 dp f .

So dp f is invertible.

Theorem 3.9 (Manifold version of the inverse function theorem).Let f : M→ M be smooth, p ∈ M with dp f : M→ M invertible. Then there areopen neighborhoods U ⊂ M of p and V ⊂ M of f (p) such that f |U : U → Vis a diffeomorphism.

Proof. The theorem is a reformulation of the inverse function theorem.

Definition 3.10 (Submersion). A map g : M→ M is called a submersion ifdpg is injective for all p ∈ M.

30

The Tangent Bundle

Definition 3.11 (Immersion). A map ψ : M → M is called an immersion ifdpψ is surjective for all p ∈ M.

Remark 3.12. Surjectivity/injectivity can be verified via linear indepen-dence, for instance by det, of rows/columns of parts of the derivative.As det is smooth, this means that if a map h : M → M is a submer-sion/immersion for some p ∈ M, then it is one in an open neighborhoodof p.

Theorem 3.13 (Submersion theorem). Let f : M → M be a submersion, i.e.for each p ∈ M the derivative dp f : TpM→ T f (p)M is surjective. Let q = f (p)be fixed. Then

M := f−1(q)is an n-dimensional submanifold of M, where n = dim M− dim M.

Remark 3.14. The sumbersion theorem is a manifold version of the implicitfunction theorem.

Proof. Take charts and apply 2.27.

Theorem 3.15 (Immersion theorem). Let f : M → M be an immersion, i.e.for every p ∈ M the differential dp f : TpM → T f (p)M is injective. Then foreach p ∈ M there is an open set U ⊂ M with p ∈ M such that f (U) is asubmanifold of M.

Proof. Take charts and apply 2.27.

Is there a global version, i.e. without passing to U ⊂ M? Assuming that fis injective is not enough.

31

3.2 The tangent bundle as a smooth vector bundle

Figure 3.1: Example of an injective immersion that is no submanifold.

Theorem 3.16. Let M be a compact manifold and f : M→ M an injective im-mersion. Then f (M) is a submanifold and f : M→ f (M) is a diffeomorphism,i.e. f is an embedding.

Remark 3.17. This time, there is no reason to state the definitions for thecomplex case again. All definitions and theorems extend to the complexcase by replacing smooth by holomorphic.

3.2 The tangent bundle as a smooth vector bun-dle

Let M be a smooth n-manifold, p ∈ M. The tangent space at p is an n-dimensional subspace of (C ∞(M))∗ given by

TpM = X | ∃γ : (−ε, ε)→ M, γ(0) = p, X f = ( f γ)′(0), ∀ f ∈ C ∞(M)

The tangent bundle is then the set

TM =⊔

p∈MTpM

and comes with a projection

π : TM→ M, TpM 3 X 7→ p ∈ M.

The set π−1(p) = TpM is called the fiber of the tangent bundle at p.

Goal: We want to make TM into a 2n-dimensional manifold.

If ϕ = (x1, . . . , xn) be a chart of M defined on U 3 p. Then we have a basis∂

∂x1

∣∣∣p

, . . . , ∂∂xn

∣∣∣p

of TpM. So there are unique y1(X), . . . , yn(X) ∈ R such that

X =∑

yi(X)∂

∂xi

∣∣∣∣∣p

.

Let (Uα, ϕα)α∈A be a smooth atlas of M. For each α ∈ A we get an openset Uα := π−1(Uα) and a function yα : Uα → Rn which maps a given vectorto the coordinates yα = (yα,1, . . . , yα,n) with respect to the frame defined byϕα.

32

The Tangent Bundle

Now, we defineϕα : π−1(U)→ Rn ×Rn = R2n

byϕα = (ϕα π, yα).

For any two charts we have a transition map φαβ : ϕα(Uα ∩Uβ)→ ϕβ(Uα ∩Uβ)

such that ϕβ

∣∣∣Uα ∩Uβ

= φαβ ϕα|Uα ∩Uβ. The chain rule yields:

yβ(X) = φ′αβ(ϕα(π(X)))yα(X).

Hence we see that ϕβ ϕ−1α is a diffeomorphism.

Topology on TM:

OTM :=¶

W ⊂ TM | ϕα(W ∩ Uα) ∈ OR2n for all α ∈ A©

.

Exercise 3.18.

a) This defines a topology on TM.

b) With this topology TM is Hausdorff and 2nd-countable.

c) All ϕα are homeomrophisms onto their image.

Because coordinate changes are smooth, this turns TM into a smooth 2n-dimensional manifold.

3.3 Vector Bundles

Now that we have learned about the tangent bundle we will see that it isonly a special case of a much more general concept, namely vector bun-dles. We will mainly consider complex one-dimensional vector bundles,also called complex line bundles, throughout this course. But first thingsfirst, so we give the following definition.

33

3.3 Vector Bundles

Definition 3.19 (Vector bundle). A smooth vector bundle of rank k is a triple(E, M, π) where E and M are manifolds and π : E→ M is smooth such that

(i) The fiber Ep := π−1 (p) has the structure of a k-dimensional realvector space.

(ii) There is U ⊂ M open, p ∈ U and a diffeomorphism φ : π−1(U) →U ×Rk such that π1 φ = π|U where π1 is the projection on the firstcomponent, i.e. for each q ∈ U the map φq : Eq → Rk is a vector spaceisomorphism defined by

Äq, φq(ψ)

ä= φ(ψ).

Definition 3.20 (Section). A smooth map ψ : M→ E is called a section of Eif π ψ = idM.

We denot the set of all smooth sections of E by

Γ(E) := ψ | ψ is a section of E

Note that Γ(E) is a real vector space under pointwise addition. In factΓ(E) is a module over the ring C∞(M): for f ∈ C∞(M), ψ ∈ Γ(E) definef ψ ∈ Γ(E) by

( f ψ)p := f (p)ψp.

Example 3.21.

(i) (Vector fields:)We have seen that the tangent bundle TM of a smooth manifold is avector bundle of rank dim M. Its smooth sections were called vectorfields.

34

The Tangent Bundle

(ii) (The trivial bundle:)The product M×Rk is called the trivial bundle of rank k. Its smoothsections can be identified with Rk-valued functions. More precisely, ifπ2 : M×Rk → Rk, then

Γ(M×Rk) 3 ψ = (p, f (p)) ←→ f := π2 ψ ∈ C ∞(M).

From now on we will keep this identification in mind and often con-fuse Γ(M×Rk) with C∞(M, Rk).

(iii) T∗M := (TM)∗ is called the cotangent bundle.

(iv) Bundles of multilinear forms with all the E1, . . . , Er, F copies of TM,T∗M or M×R are called tensor bundles. Sections of such bundles arecalled tensor fields.

Ways to make new vector bundles out of old ones

The general principle is the following:Any linear algebra operation that gives new vector spaces out of given onescan be applied to vector bundles over the same base manifold.

Example 3.22.Let E be a rank k vector bundle over M and F be a rank ` vector bundleover M.

(i) Then E⊕ F denotes the rank k + ` vector bundle over M the fibers ofwhich are given by (E⊕ F)p = Ep ⊕ Fp.

(ii) Then Hom(E, F) denotes the rank k · ` vector bundle over M with fibergiven by Hom(E, F)p := f : Ep → Fp | f linear.

(iii) E∗ = Hom(E, M×R) with fibers (E∗)p = (Ep)∗.

Let E1, . . . .Er, F be vector bundles over M.

(iv) Then a there is new vector bundle E∗1 ⊗ · · · ⊗ E∗r ⊗ F of rank

rankE1 · · · rankEr · rankF

with fiber at p given by

E∗1p ⊗ · · · ⊗ E∗rp ⊗ Fp = β : E1p × . . .× Erp → Fp | β multilinear.

35

3.3 Vector Bundles

Definition 3.23 (Vector bundle isomorphism). Two vector bundles E →M, E→ M are called isomorphic if there is a bundle isomorphism

f : E→ E ∈ Γ Hom(E, F),

that means π f = π (fibers to fibers) and f |Ep: Ep → Ep is a vector space

isomorphism.

Definition 3.24 (trivial vector bundle). A vector bundle E → M of rank kis called trivial if it is isomorphic to the trivial bundle M×Rk.

Remark 3.25. If E→ M is a vector bundle of rank k then, by definition, eachpoint p ∈ M has an open neighborhood U such that the restricted bundleE|U := π−1(U) is trivial, i.e. each bundle is locally trivial.

Definition 3.26 (Frame field). Let E → M be a rank k vector bundle,ϕ1, . . . , ϕk ∈ Γ(E). Then (ϕ1, . . . , ϕk) is called a frame field if for each p ∈ Mthe vectors ϕ1(p), . . . , ϕk(p) ∈ Ep form a basis.

Proposition 3.27. A vector bundle E is trivial if and only if it has a frame field.

Proof.

”⇒”: Let E be trivial then there ∃F ∈ ΓHom(E, M×Rk) such that

Fp : Ep → p ×Rk

is a vector space isomorphism for each p. Then, for i = 1, . . . , k defineϕi ∈ Γ(E) by ϕp := F−1(p × ei).

”⇐”: Let (ϕ1, . . . , ϕk) be a frame field of E, then we can define a bundleisomorphism F ∈ ΓHom(E, M×Rk) as the unique map such that

Fp(ϕi(p)) = p × ei

for each p ∈ M.

Example 3.28. A rank 1 vector bundle E (a line bundle) is trivial if and onlyif there exists a nowhere vanishing ϕ ∈ Γ(E).

36

The Tangent Bundle

Example 3.29.

(i) Consider E = S1 × R. Then we can define a section ψ ∈ Γ(E) bysetting ψp = (p, 1) for all p ∈ S1. As R is only one-dimensional, ψ, asit is nowhere vanishing, defines a frame field of E, thus E is trivial.

(ii) The Mobius band shaped rank 1 vector bundle over S1 below is nottrivial, as any non-vanishing (i.e. non-zero) section in Γ(E) cannot besmooth.

(iii) There is a nowhere vanishing vector field X ∈ Γ(TM) for M = T2.

Example 3.30. Let M ⊂ R` be a submanifold of dimension n. Then thisadmits a rank `− n vector bundle NM (the normal bundle of M). It is givenby

NpM = (NM) = (TpM)⊥ ⊂ TpR` = p ×R`.

As a matter of fact, The normal bundle of a Moebius band is not trivial.

Example 3.31. The tangent bundle of S2 is not trivial - a fact known as thehairy ball theorem: Every vector field X ∈ Γ(T S2) has zeros.

37

3.4 The Lie-bracket of Vector Fields

3.4 The Lie-bracket of Vector Fields

Let X ∈ Γ(TM), f ∈ C∞(M). Then

X f : M→ R, p 7→ Xp f ,

is smooth. So X can be viewed as a linear map

C∞(M)→ C∞(M), f 7→ X f .

Theorem 3.32 (Leibniz’s rule). Let f , g ∈ C∞(M), X ∈ Γ(TM), then

X( f g) = (X f )g + f (Xg).

Definition 3.33 (Lie algebra). A Lie algebra is a vector space g together witha skew bilinear map [., .] : g× g→ g which satisfies the Jacobi identity,

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0.

Theorem 3.34 (Lie algebra of endomorphisms). Let V be a vector space.End(V) together with the commutator [., .] : End(V)× End(V) → End(V),[A, B] := AB− BA forms a Lie algebra.

Proof. Certainly the commutaor is a skew bilinear map. Further,

[A, [B, C]] + [B, [C, A]] + [C, [A, B]]= A(BC− CB)− (BC− CB)A + B(CA− AC)− (CA− AC)B + C(AB− BA)− (AB− BA)C,

which is zero since each term appears twice but with opposite sign.

Theorem 3.35. For all f , g ∈ C∞(M), X, Y ∈ Γ(M),

[ f X, gY] = f g[X, Y] + f (Xg)Y− g(Y f )X.

Lemma 3.36 (Schwarz lemma). Let ϕ = (x1, . . . , xn) be a coordinate chart.

Then [∂

∂xi,

∂

∂xj] = 0.

Exercise 3.37. Prove Schwarz lemma above.

38

The Tangent Bundle

Theorem 3.38. Γ(TM) ⊂ End(C∞(M)) is a Lie subalgebra.

This theorem can alternatively be stated in an equivalent way, namely

Theorem 3.39. For X, Y ∈ Γ(TM) there is a unique vector field [X, Y] ∈Γ(TM) such that for all f ∈ C∞(M)

[X, F] f = x(Y f )−Y(X f ).

Proof. The right hand side can be evaluated at p ∈ U, U ⊂ M open, thus

only depends on X|U, Y|U, f |U. So if X =∑

iai

∂

∂xiand Y =

∑j

bj∂

∂xj, we

get

X(Y f )−Y(X f )

=n∑

i=1ai

∂

∂xi

Ñn∑

j=1bj

∂ f∂xj

é−

n∑i=1

bi∂

∂xi

Ñn∑

j=1aj

∂ f∂xj

é=

n∑i=1

Ñn∑

j=1

Çai

∂bj

∂xi− bi

∂aj

∂xi

åé∂ f∂xj

+n∑

i,j=1aibj

(∂2 f

∂xi∂xj− ∂2 f

∂xj∂xi

)

=

Ñn∑

i=1

Ñn∑

j=1

Çai

∂bj

∂xi− bi

∂aj

∂xi

åé∂

∂xj

éf

The last equality relies on the fact that the second sum is zero, as by lemma3.36 partial derivatives commute. By defining

[X, Y] :=n∑

i=1

Ñn∑

j=1

Çai

∂bj

∂xi− bi

∂aj

∂xi

åé∂

∂xj

we yield a unique [X, Y] ∈ Γ(TM).

39

4. Almost Complex Structures

4.1 Tensors

Let E1 . . . , El, F be vector bundles over the same manifold M. Further let Tbe the vector bundle over M with fibers

Tp =¶

µp : (E1)p × . . .× (El)p → F | µp multilinear©

Consider µ ∈ Γ(T), then one can define the following:

µ : Γ(E1)× . . .× Γ(El)→ Γ(F)

byµ(ψ1, . . . , ψl)p = µp(ψ1(p), . . . , ψl(p)) ∈ Fp

To first define µ is often is the most convenient way to define µ, but it isnot clear yet under which circumstances such a µ really comes from anµ ∈ Γ(T), as most of them don’t.A standard way to check if this is the case is provided by the followingtheorem.

Theorem 4.1. Let E1 . . . , El, F be vector bundles over M and let µ : Γ(E1)×. . .× Γ(El) → Γ(F) be real linear in all its arguments. Then the following areequivalent:

(i) For all f ∈ C∞(M), ψ1 ∈ Γ(E1),...,ψl ∈ Γ(El), µ is tensorial in allarguments, i.e.

µ( f ψ1, . . . , ψl) = . . . = µ(ψ1, . . . , f ψl) = f µ(ψ1, . . . , ψl)

(ii) There is µ ∈ Γ(T) such that µ = µ.

Proof. We only check this for l = 1, i.e. T = Hom(E, F).

40

Almost Complex Structures

”⇐” If µ(ψ)p = µp(ψp) then, because

µ( f ψ)p = µp( f (p)ψp) = f (p)µp(ψp) = ( f µ(ψ))p

it certainly holds that tildeµ( f ψ) = f µ(ψ).

”⇒” Assume that µ( f ψ) = f µ(ψ) for all f ∈ C∞(M), ψ ∈ Γ(E). Let p ∈ Mthen we claim that µp(ψ) = (µ(ψ))p only depends on ψp.Choose a frame field ϕ1, . . . , ϕl ∈ Γ(E|U) for some U ⊂ M open withp ∈ U. Further choose a bump-function f ∈ C∞(M) such that f (q) = 0 , q /∈ U

f (q) = 1 , q ∈ V ⊂ U

V open.We want to show that ψp = ψp implies that (µ(ψ))p = (µ(ψ)p, or inother words, considering a section ψ := ψ− ψ, that

ψp = 0 ⇒ (µ(ψ))p = 0

then by the linearity of µ the claim follows.For an easier notation we will only write ψ instead of ψ.So let ψ ∈ Γ(E) with ψp = 0, then locally on U we can express ψ viathe frame field as

ψ|U = a1ϕ1 + . . . + ak ϕk

with a1, . . . , ak ∈ C∞(U). The trick is now, using the tensoriality of ψto compute

f 2(µ(ψ)) = µ( f 2ψ)

= µ (( f a1)( f ϕ1) + . . . + ( f ak)( f ϕk))

= ( f a1)µ( f ϕ1) + . . . + ( f ak)µ( f ϕk)

where we used that ( f aj) ∈ C∞(M) and ( f ϕj) ∈ Γ(E) for all j =1, . . . , k, thus are defined on the whole of M as the bump functionassures that they vanish outside of U.Evaluating at p yields, as a1(p) = . . . = ak(p) = 0 because ψp = 0,

f (p)2 (µ(ψ))p = (µ(ψ))p = 0

where we also used that by construction f (p) = 1. By rememberingthat we denote ψ by ψ the claim follows.

Remark 4.2. In the following we keep this identification be tensors andtensorial maps in mind and just speak of tensors.

41

4.2 The Newlander-Nirenberg Theorem

4.2 The Newlander-Nirenberg Theorem

Definition 4.3 (Almost complex structure). An almost complex structureon a manifold M is a section J ∈ ΓEnd(TM) with J2 = −Id.

Definition 4.4 (Almost complex manifold). An almost complex manifoldis a manifold together with an almost complex structure.

Remark 4.5.

(i) If M is an almost complex manifold, then all tangent spaces TpM arecomplex vector spaces. This means dim M = 2n for some n ∈N.

(ii) All complex manifolds are almost complex. We only need to definefor X ∈ TpM, p ∈ Uα for a chart (Uα, ϕα) in the atlas

JX = dϕ−1(JCdϕX)

This is well-defined because coordinate changes are biholomorphic.A more detailed discussion can be found in the Appendix.

(iii) Not every almost complex manifold is a complex manifold. As aspecial case for C2 this is true, but not in general. We will soon learnabout a criterion to check this.

Definition 4.6 (Nijenhuis-tensor). Let M be a manifold, A ∈ ΓEnd(TM).Then the Nijenhuis tensor NA is defined on vector fields X, Y ∈ Γ(TM) as

NA(X, Y) := −A2[X, Y]− [AX, AY] + A([AX, Y] + [X, AY]).

We have to check that NA really is tensorial:First NA is skew in X and Y, thus it is enough to check that it is tensorialin the first slot:

NA( f X, Y) = f NA(X, Y) + A2((Y f )X) + ((AY) f )AX− A((Y f )AX + A((Y) f AX + ((AY) f )X)

= f NA(X, Y).

where, for the last equality, we used that A is linar.

Example 4.7. Let M = Cn and JX = iX then NJ = 0. This can be seenas follows. As NJ is tensorial, we can without loss of generality choosevector fields X, Y that are constant. Then also JX, JY are constant and allLie-brackets vanish.

42


Theorem 4.8 (Newlander–Nirenberg). An almost complex manifold (M, J)is complex if and only if its Nijenhuis tensor vanishes, NJ = 0.

Proof. One direction follows directly from the previous example. The otherdirection (”⇐”) is hard and far beyond the scope of this course.

4.3 Connections on Vector Bundles

If we consider a rank k vector bundle E → M, then a section ψ ∈ Γ(E) issimilar to a function ψ : M→ Rk, only that ψ(p) ∈ Ep ∼= Rk.Now it is only natural that we want to think about how a section changes inE when we vary the point p ∈ M.

The problem that arises with this is that two tangent vectors ψp, ψq ∈ E withdifferent basepoints p and q do, in the first place, have nothing in commonas they lie in distinguished vector spaces Ep and Eq.

We will introduce a suitable substitute that will provide remedy in the casethat we have a path in M connecting p and q.

Definition 4.9 (Connection). A connection on a vector bundle E → M is abilinear map

∇ : Γ(TM)× Γ(E)→ Γ(E)

such that for all f ∈ C ∞(M), X ∈ Γ(TM), ψ ∈ Γ(E),

(i) ∇ f Xψ = f∇Xψ

(ii) ∇X f ψ = (X f )ψ + f∇Xψ.

The proof of the following theorem will be postponed until we have estab-lished the existence of a so called partition of unity.

43


Theorem 4.10. On every vector bundle E there is a connection ∇.

Definition 4.11 (Parallel section).Let E → M be a vector bundle with connection ∇. Then ψ ∈ Γ(E) is calledparallel if, for all X ∈ TM,

∇Xψ = 0

Example 4.12. As sections are similar to functions, the main example willbe E = M×Rk. Then for a section ψ ∈ Γ(E),

ψp =Ä

p, ψ(p)ä

for ψ : M→ Rk.

Then naturally we define

(∇Xψ)p :=Ä

p, Xpψä=Ä

p, dpψ(X)ä

.

This particular connection is called the trivial connection on the trivial bundleand thus is often also denoted by d, so that

∇Xψ = dXψ.

We already know, that there is a connection on every vector bundle, soquickly the question of how many there are arises. We will later see, thatthere is in fact a unique connection that has particulary nice properties, butthe following theorem provides that in general a connection is not unique.

Definition 4.13 (Difference tensor). For two connections ∇, ∇ on a vectorbundle E→ M then the map

A : Γ(TM)× Γ(E)→ Γ(E), AXψ = ∇Xψ−∇Xψ

is called the difference tensor of ∇ and ∇.

To make sure that the difference tensor is well defined, we need to checkthat it is tensorial in both arguments, then the claim follows by theorem4.1.It is by the properties of a connection

A f Xψ = ∇ f X −∇ f X = f ∇Xψ− f∇Xψ = f AXψ

AX ( f ψ) = ∇( f ψ)−∇X( f ψ) = (X f )ψ + f ∇Xψ− (X f )ψ− f∇Xψ = f AXψ

thus A is well defined.

44


Such an A is more or less the same thing (i.e. isomorphic) as a section,meaning

A ∈ Γ Hom (TM, End(E)) .

Thus given any connection ∇ on E, all other connections on E are of theform

∇ = ∇+ A.

In particular: On the the trivial bundle M ×Rk all connections are of theform

∇ = d + A.

In many applications vector bundles E have additional structures and usu-ally connections on E are required to be compatible with these. What thisis supposed to mean will be clarified in the following.

Definition 4.14 (Complex connection). Let E be a vector bundle with com-plex structure J ∈ End(E) on E, then (E, J) is a complex vector bundle. Aconnection ∇ on E is called complex connection if it is compatible with J, i.e.for all X ∈ Γ(TM), ψ ∈ Γ(E)

∇X(Jψ) = J∇Xψ.

A complex connection ∇ is thus complex linear in ψ.

Let E → M be a rank k vector bundle and Sym(E) be the bundle whosefiber at p ∈ M consists of all symmetric bilinear forms Ep × Ep → R. Wesee that

rank(Sym(E)) = k(k+1)2 .

Definition 4.15 (Metric). A metric on E is a section 〈., .〉 of Sym(E) suchthat 〈., .〉p is a Euclidean inner product for all p ∈ M, i.e. 〈., .〉p is positivedefinite for all p ∈ M.

Definition 4.16 (Euclidean vector bundle). A vector bundle together witha metric (E, 〈., .〉) is called Euclidean vector bundle.

Definition 4.17 (Riemannian manifold). A Riemannian manifold is a man-ifold M together with a Riemannian metric, i.e. a metric 〈., .〉 on TM.

45


Definition 4.18 (Metric connection). Let (E, 〈., .〉) be a Euclidean vectorbundle over M. Then a connection ∇ is called metric if for all ψ, ϕ ∈ Γ(E)and X ∈ Γ(TM) we have

X〈ψ, ϕ〉 = 〈∇Xψ, ϕ〉+ 〈ψ,∇X ϕ〉.

Putting these two notions together we get a sensible definition of the com-patibility of complex structures and metrics on vector bundles.

Definition 4.19 (Compatibility of J and 〈., .〉). A euclidean inner product〈., .〉 and a complex structure J on a vector bundle E are said to be compatibleif for all u, v ∈ V

〈Ju, v〉 = − 〈u, JV〉or equivalently if J∗ = −J, i.e. J is skew-adjoint.

Remark 4.20. If a metric 〈., .〉 is compatible with the complex structure J ona vector bundle E, then J is automatically orhtogonal, as it holds that

〈Ju, Jv〉 = −¨u, J2v

∂= 〈u, v〉 .

Definition 4.21 (Complex scalar product). We define the complex scalarproduct by

〈., .〉C : V ×V → C, 〈u, v〉C = 〈u, v〉+ i 〈Ju, v〉 .

We see that by definition the following two equalities hold:

〈Ju, v〉C = 〈Ju, v〉 − i 〈u, v〉 = −i 〈u, v〉C

〈u, Jv〉C = 〈u, Jv〉+ i 〈Ju, Jv〉 = 〈u, Jv〉+ i 〈u, v〉 = i 〈u, v〉CSo 〈., .〉C is anti-linear in the left argument and linear in the right argument,thus hermitian.If conversely 〈., .〉C is hermitian, then 〈., .〉 := Re 〈., .〉C is compatible with J.

Definition 4.22 (unitary vector bundle). A unitary vector bundle is a com-plex vector bundle together with a compatible euclidean metric 〈., .〉.

46


Definition 4.23 (Compatible ∇ for unitary bundles). A connection ∇ ona unitary vector bundle is called compatible if it is complex and metric.

Let us denote the set of sections in endomorphisms on E that commutewith a given complex structure J on E by

Γ End+(E).

The set of sections in endomorphisms on E that anti-commute with a givencomplex structure J on E is denoted by

Γ End−(E).

Proposition 4.24. If ∇ is a connection on a vector bundle E that is compatiblewith J, then another connection ∇ = ∇+ A on E is compatible with J if andonly if

AX J = JAX

this means AX ∈ Γ End+(E).

Proof. We have

J∇ψ = ∇X(Jψ)

⇔ J∇Xψ + JAXψ = ∇X(Jψ) + AX Jψ

⇔ J∇Xψ + JAXψ = J∇X(ψ) + AX Jψ

⇔ JAXψ = AX Jψ

Proposition 4.25. If ∇ is a connection on a vector bundle E that is compatiblewith 〈., .〉, then another connection ∇ = ∇+ A on E is also compatible with〈., .〉.

Proof. Again we use the definition and derive that

X 〈ψ, φ〉 =¨∇Xψ, φ

∂+¨ψ, ∇Xφ

∂⇔ 〈∇Xψ, φ〉+ 〈ψ,∇Xφ〉 = 〈∇Xψ + AXψ, φ〉+ 〈ψ,∇Xφ + AXφ〉

⇔ 0 = 〈AXψ, φ〉+ 〈ψ, AXφ〉⇔ A∗X = −AX

which is a true statement.

When is a connection ∇ on TM compatible with the status of TM as thetangent bundle?

47


Definition 4.26 (affine connection). A connection ∇ on the tangent bundleTM is called an affine connection.

Definition 4.27 (Torsion tensor). Let ∇ be an affine connection, then themap

T : Γ(TM)× Γ(TM)→ Γ(TM), (X, Y) 7→ ∇XY−∇YX− [X, Y]

is called the torsion tensor of ∇.

To assure that T is well defined, we need to check the tensoriality of it.As T is obviously skew it suffices to check that for the left argument. Forf ∈ C∞(M) and X, Y ∈ Γ(TM) we have

T( f X, Y) = f∇XY− (Y f )X− f∇XY + (Y f )X = f T(X, Y)

thus T really is a tensor.

Example 4.28. Take a look at our formula for the Lie-brackets expressed inlocal coordinates. We see that the trivial connection ∇ on TRn has T = 0.

Definition 4.29 (Torsion free). A connection ∇ in TM is called torsion-freeif the torsion tensor vanishes, i.e. for all X, Y ∈ Γ(TM)

∇XY−∇YX = [X, Y].

So we have now introduced a lot of structures on a vector bundle E, butnot every bundle admits all of these, for example there exists no complexstructure J on TS4. But we already learned, that every vector bundle has aRiemannian metric. Further, we say a complex structure on E is compatiblewith the metric 〈., .〉 if

〈u, v〉 = 〈Ju, Jv〉⇔ J∗ = −J⇔ J∗ J = I

So is it true, that if we have a vector bundle with both, a complex struc-ture J and a metric 〈., .〉, we can always have them in a way that they arecompatible?

Theorem 4.30. Every vector bundle E with a complex structure J has a com-patible metric 〈., .〉.

48


Proof. Choose 〈., .〉∼ to be any metric on E and define

〈ψ, φ〉 := 12 (〈ψ, φ〉∼ + 〈Jψ, Jφ〉∼)

then〈Jψ, φ〉 = 1

2 (〈Jψ, φ〉∼ − 〈Jψ, Jφ〉∼) = − 〈ψ, Jφ〉where we used that J∗ = −J.

Although we did not actually prove it, we will use theorem 4.8 to give aproof of the following theorem. In one part of the proof we will also makeuse of the partition of unity, of which we will learn in the next chapter.

Theorem 4.31. An almost complex manifold is complex if and only if there is atorsion-free connection ∇ on the tangent bundle TM that is compatible with J,i.e. ∇J = 0.

Proof.

”⇒”: It is

NJ(X, Y) = [X, Y]− [JX, JY] + J ([JX, Y] + [X, JY])= ∇XY−∇YX− J∇JXY + J∇JYX

+ JÄ∇JXY− J∇Y(X) + J∇XY−∇JYX

ä= 0

”⇐”: Let (M, J) be complex, then using charts (Uα, ϕα), we locally findcompatible connections∇α. Using a partition of unity, we ”glue” themtogether to obtain the desired connection.

Theorem 4.32 (Fundamental theorem of Riemannian geometry). On aRiemannian manifold there is a unique affine connection ∇ which is both metricand torsion-free. ∇ is called the Levi-Civita connection.

Proof. Uniqueness: Let ∇ be metric and torsion-free, X, Y, Z ∈ Γ(TM).Then

X〈Y, Z〉+ Y〈Z, X〉 − Z〈X, Y〉= 〈∇XY, Z〉+ 〈Y,∇XZ〉+ 〈∇YZ, X〉+ 〈Z,∇YX〉 − 〈∇ZX, Y〉 − 〈X,∇ZY〉

= 〈∇XY +∇YX, Z〉+ 〈Y,∇XZ−∇ZX〉+ 〈∇YZ−∇ZY, X〉= 〈2∇XY− [X, Y], Z〉+ 〈Y, [X, Z]〉+ 〈[Y, Z], X〉.

49


Hence we obtain the so called Koszul formula:

〈∇XY, Z〉 = 12

ÄX〈Y, Z〉+ Y〈Z, X〉

− Z〈X, Y〉+ 〈[X, Y], Z〉 − 〈Y, [X, Z]〉 − 〈[Y, Z], X〉ä.

So ∇ is unique. Conversely define ∇XY by the Koszul formula (for this tomake sense we need to check tensoriality). Then check that this defines ametric torsion-free connection.

Definition 4.33 (Almost hermitian manifold). An almost complex mani-fold (M, J) together with a metric on TM that is compatible with the almostcomplex structure J on TM is called an almost hermitian manifold.

Remark 4.34. Almost hermitian manifold are sometimes also called unitarymanifolds.

Definition 4.35 (Kahler–manifold). An almost hermitian manifold for whichthe Levi-Civita connection ∇ of 〈., .〉 is compatible with the almost complexstructure J is called a Kahler-manifold.

Example 4.36.

(i) On CPn (which is a complex manifold) there is the Fubini-Study-metric〈., .〉 which turns CPn into a Kahler-manifold.

(ii) Let M be a complex manifold and M ⊂ M a complex submanifold,then M has an induced J which is also complex.

(iii) Let M be a Riemannian manifold and M ⊂ M a submanifold, then Mhas an induced metric 〈., .〉.

Exercise 4.37. Show that a complex submanifold of a Kahler-manifold isKahler

Theorem 4.38 (A Version of Kodaira embedding theorem). Every compactKahler manifold admits a complex embedding (not necessarily isometric) in someCPn.

Proof. The proof is way to difficult to do it in the scope of this lecture, sowe will skip it. It may be seen in a lecture about algebraic geometry, as theresult is one of the highlights of that lecture.

50


ConsiderM := CPn \ 0z 7→ 2z

∼= S1 × S1.

M is called a Hopf -manifold and is a complex manifold.

Theorem 4.39. The Hopf-manifold (M, J) mentioned above as a compex mani-fold has no Kahler-metric and therefore cannot be realized in any CPn.

Proof. This proof is not too hard. It may be done in the end of this course,when we established more theory to make use of.

4.4 Partition of unity

Theorem 4.40 (partition of unity). Let M be a manifold, A ⊂ M compact and(Uα)α∈I an open cover of A. Then there are $1, . . . , $m ∈ C∞(M) such that foreach i ∈ 1, . . . , m there is αi ∈ I such that supp $i ⊂ Uαi is compact. Moreover

$i(p) ≥ 0 for all p ∈ M andm∑

i=0$i(p) = 1 for all p ∈ A.

Proof. We already know that there is a function g ∈ C∞(Rn) such thatg(p) ≥ 0 for all p ∈ Rn and g(p) > 0 if p ∈ D := x ∈ Rn||x| < 1. Define

Dp := x ∈ Rn||x− p| < 1, D := x ∈ Rn||x| < 2,

Dp := x ∈ Rn||x− p| < 2.

For each p ∈ A there is a chart (Up, ϕp) such that ϕp : Up → Dp is adiffeomorphism and Up ⊂ Uα for some α ∈ I.If we define

Vp := ϕ−1p (Dp)

then (Vp)p∈A is an open cover of A. This implies that there are p1, . . . , pm ∈A such that A ⊂ Vp1 ∪ · · · ∪Vpm .

Now define for i ∈ 1, . . . , m $i : M→ R by $i =

0 if p /∈ Vpi

g ϕpi if p ∈ Vpi

This means ($1 + · · · + $m)(p) ≥ 0 always holds, and in particular ($1 +· · ·+ $m)(p) > 0 for p ∈ A. If now A = M, then we define

$i :=$i

($1 + · · ·+ $m)

and we are done.Otherwise, since A is compact, $1 + · · ·+ $m attains its minimum in A, i.e.there is an ε > 0 such that ($1 + · · ·+ $m)(p) ≥ ε for all p ∈ A. Consturct

51

4.4 Partition of unity

h ∈ C∞(M) such that h(x) > 0 for all x ∈ R and h(x) = x for x ≥ ε. Nowfor i ∈ 1, . . . , m define

$i :=$i

h($1 + · · ·+ $m)

then clearly ($1 + · · ·+ $m)|A = 1.

Theorem 4.41 (partition of unity - general version). Let M be a manifoldand (Uα)α∈I an open cover of M, i.e. ∪α∈IUα = M. Then there is a family($β)β∈J with $β ∈ C∞(M) such that

1. for each β ∈ J there is α ∈ I such that supp $β ⊂ Uα is compact.

2. $i(p) ≥ 0 for all p ∈ M.

3. $β(p) 6= 0 only for finitely many β ∈ J and and∑β∈J

$β(p) = 1 for all

p ∈ M.

Proof. This theorem will remain without proof, as we only cite it to em-phasize the existence of a more general version, but do not actually use itmuch.

Nonetheless we will shortly give some applications of the general partitionof unity.

Theorem 4.42. Every manifold has a Riemannian metric.

Proof. We have coordinate charts (Uα, ϕα), i.e. an open cover (Uα)α∈I ofM and a Riemannian metric gα on Uα, where gα = ϕ∗αgRn is the pullbackmetric1. Now we choose a partition of unity ($β)β∈J subordinate to thecover (Uα)α∈I .Define

g :=∑β∈J

$βgα(β)

which is, as a linear combination of positive definite symmetric bilinearforms with positive coefficients, also positive definite. Then g is a Rieman-nian metric on M.

Remark 4.43. It is not in general true that every manifold has a Pseudo-Riemannian metric, i.e. there is no Lorenz-metric on S2.

1The concept of pullbacks is introduced in differential geometry. We add a short intro-duction on these in the appendix.

52


Theorem 4.44. Every vector bundle E over M has a connection.

Remark 4.45. If∇1, . . . ,∇m are connections on any bundle E and λ1, . . . , λm ∈C∞(M) with λ1 + · · ·+ λm = 1, then ∇ defined by

∇Xψ := λ1∇1Xψ + · · ·+ λm∇m

Xψ

for X ∈ Γ(TM), ψ ∈ Γ(E), certainly is a connection.

Proof. Locally (on Uα) E looks like Uα ×Rk, therefore, by pulling pack thetrivial connection, we have a connection ∇α on E|Uα . Choose a partition ofunity ($β)β∈J subordinate to the cover (Uα)α∈I , then we can say

∇ =∑β∈J

$β∇α(β)

is certainly a connection, as only finitely many of the $ 6= 0.

4.5 Conformal Equivalence

Let (M, 〈., .〉) be a Riemannian manifold and γ : [0, 1]→ M a smooth curveon M, then the length of γ is given by

L(γ) :=∫ 1

0

∣∣∣γ′∣∣∣.

If we consider another smooth curve η : [0, 1] → M on M such that γ(s) =η(t) for some s, t ∈ [0, 1] then

cos α :=〈γ′(s), η′(t)〉|γ′(s)||η′(t)|

where α is the angle between γ and η at p := γ(s) = η(t). This means thata Riemannian metric allows us to measure angles on a manifold M.

53

4.5 Conformal Equivalence

Definition 4.46 (Conformal equivalence). Two Riemannian metrics 〈., .〉and 〈., .〉∼ on a manifold M are called conformally equivalent if there is func-tion u ∈ C∞(M) such that

〈., .〉∼ = e2u 〈., .〉 .

We see that two Riemannian metrics are conformally equivalent if they onlydiffer by a positive scaling factor for all p ∈ M.

Remark 4.47. Conformal equivalence defines an equivalence relation on theset of Riemannian metrics on M.

Definition 4.48 (Conformal structure). A conformal structure on a mani-fold M is an equivalence class of conformally equivalent Riemannian metricson M.

A question that immediately arises is that if we know how to make con-formal changes of a metric, do we also know how the corresponding Levi-Civita connection on M changes? Luckily we do, as the following theoremassures.

Theorem 4.49. Let 〈., .〉∼ = e2u 〈., .〉 then the corresponding Levi-Civita con-nections ∇, ∇ are related as

∇XY = ∇XY + 〈X, G〉Y + 〈Y, G〉X− 〈X, Y〉G

where G ∈ Γ(TM) is defined as G = grad u with respect to 〈., .〉, i.e. 〈G, Z〉 =du(Z) = Zu.

Proof. It is easy to see that ∇ really defines a connection, so we have toproof that it is torison free and metric with respect to 〈., .〉∼. We have, forX, Y, Z ∈ Γ(TM)

∇XY− ∇YX = ∇XY + 〈X, G〉Y + 〈Y, G〉X− 〈X, Y〉G−∇YX− 〈X, G〉Y− 〈Y, G〉X + 〈X, Y〉G

= ∇XY−∇YX= [X, Y]

thus ∇ is torsion free. Further, on the one hand

X 〈Y, Z〉∼ = XÄe2u 〈Y, Z〉

ä= e2u (2(Xu) 〈Y, Z〉+ 〈∇XY, Z〉+ 〈Y,∇XZ〉)= e2u (2 〈G, X〉〈Y, Z〉+ 〈∇XY, Z〉+ 〈Y,∇XZ〉)

54


and on the other hand

e−2u Ä¨∇XY, Z∂+¨Y, ∇XZ

∂ä= 〈∇XY, Z〉+ 〈Y,∇XZ〉+ 〈X, G〉〈Y, Z〉+ 〈Y, G〉〈X, Z〉 − 〈X, Y〉〈G, Z〉+ 〈X, G〉〈Z, Y〉+ 〈Z, G〉〈X, Y〉 − 〈X, Z〉〈G, Y〉= 〈∇XY, Z〉+ 〈Y,∇XZ〉+ 2 〈G, X〉〈Y, Z〉

55

5. Integration on manifolds

5.1 Volume Forms

In the following we will naturally handle differential forms and pullbackswithout having them especially defined. A short introduction on both top-ics can be found in the appendix.

In R3 we know that for three vectors X, Y, Z ∈ R3 we have that the volumeof the parallelepiped spanned by them has volume

|det(X, Y, Z)|.

We want to introduce n-dimensional versions of this. To do so, we need thefollowing.

Definition 5.1 (Alternating k-forms). Let V be a vector space, then, fork ≥ 1 we define

Λk(V) := ω : V × . . .×V → R | ω is k-linear and alternating

i.e. for v1, . . . , vl ∈ V with i 6= j

ω(v1, . . . , vi, . . . , vj, . . . , vk) = −ω(v1, . . . , vj, . . . , vi, . . . , vk).

Consequently, for k = 0 we set

Λ0(V) = R.

56

Integration on manifolds

There are special cases that we will frequently use. The first one is

Λ1(V) = V∗

Further, if V is of dimension n ∈N then

dim Λn(V) = 1.

This is because V ∼= Rn as a vector space and on Rn we have

detRn ∈ Λn(Rn)

with detRn 6= 0, where detRnis the usual determinant function on Rn. From

linear algebra we know that detRncan be defined as the unique element of

Λn(Rn) with detRn(e1, . . . , en) = 1. In particular, if ω ∈ Λn(Rn) with ω 6= 0

thendetRn

=ω

ω(e1, . . . , en).

Definition 5.2 (Determinant form). For an n-dimensional vector space V,a non-zero element det ∈ Λk(V) is called a determinant form on V.

Definition 5.3 (Volume form). A volume form on an n-dimensional mani-fold M is a section det ∈ ΓΛn(M) such that detp 6= 0 for all p ∈ M.

Note that in the upper definition we use the notation

Λn(M) := Λn(TM)

for the vector bundle whose fibers at p ∈ M are

Λn(M)p := Λn(TpM).

We now want to introduce the notion of an orientation. For an n-dimensionalmanifold M this is often defined by the existence of a nowhere vanishingn-form on M, but to emphasize that such a particular form is more of aconsequence of an orientation we will define it in a way that allows us tohave it as a theorem.

57

5.1 Volume Forms

Definition 5.4 (Orientation).

(i) Two charts (U, ϕ), (V, ψ) of a manifold M are called consistently ori-ented if the coordinate change φ : ϕ(U ∩V)→ ψ(U ∩V) is orientationpreserving, i.e. det φ′ > 0.

(ii) An orientation on a manifold M is an atlas of consistently orientedcharts which is maximal in this sense.

(iii) A manifold M is called orientable if M has a consistently oriented atlas.

Theorem 5.5. An n-dimensional manifold M is orientable if and only if M hasa volume form ω ∈ Ωn(M).

Proof.

”⇒”: Let (Uα, ϕα)α∈I be a consistently oriented atlas. Choose a partitionof unity ($β)β∈J subordniate to the open cover (Uα)α of M. Let ϕ :=ϕα = (x1, . . . , xn), ϕ := ϕα = (y1, . . . , yn) then on Uα ∩Uα it is

ϕ = φ ϕ

Then

dϕ : TpM→ Tϕ(p), z 7→

Üdy1(z)

...dyn(z)

ê.

Thusdϕ = dφ dϕ

58


which is Üdy1(z)

...dyn(z)

ê= φ′(ϕ(p))

Üdx1(z)

...dxn(z)

êIt is left as an exercise to show that this implies dy1 ∧ . . . ∧ dyn =detÄφ′ ϕ

ädx1 ∧ . . .∧ dxn. In short, this is true for every chart and we

yielddx1 ∧ . . . ∧ dxn = ϕ∗αdetRn

dy1 ∧ . . . ∧ dyn = ϕ∗αdetRn.

Definingω :=

∑β∈J

$β ϕ∗α(β)detRn.

will then do the trick. Checking that this really defines a volume formis also left as an exercise.

”⇐”: Let ω ∈ Ωn(M) be the volume form of M For each chart (U, ϕ) ofM decompose U = U1 ∪ . . . ∪Um with Uj open and connected. AsΩn(M) is 1-dimensional it is

ϕ∗detRn= λω

with λ ∈ C∞(U). Now modify ϕ by past-composing ϕ|Uj by a reflec-tion of Rn on each Uj for which λ|Uj < 0. By doing this we get ϕ withλ > 0. Doing so for every chart in the atlas yields an atlas (Uα, ϕα)α∈Iof mutually compatible orientation.

5.2 Integration of Forms

Definition 5.6 (support of a section). If E is a vector bundle over M andψ ∈ Γ(E), then we define the support of ψ as

supp ψ := p ∈ M|ψp 6= 0

Remark 5.7. The support is well defined as p ∈ M|ψp 6= 0, as the com-plement of the closed set p ∈ M|ψp = 0, is open.

We defineΓ0(M, E) := ψ ∈ Γ(E) | supp(ψ) compact

andΩn

0(M, E) := Γ0Λk(M, E).

59

5.2 Integration of Forms

Definition 5.8 (Integral of ω ∈ Ωn0(R

n)). If ω ∈ Ωn0(R

n), then we define∫Rn

ω :=∫

Rnf .

Theorem 5.9. Given ω, ω ∈ Ωn0(R

n) and a diffeomorphism ϕ : supp ω →supp ω such that ω = ϕ∗ω and det ϕ′ > 0, then∫

Rnω =

∫Rn

ω

Proof. Let ω = f dx1 ∧ · · · ∧ dxn and ω = f dx1 ∧ · · · ∧ dxn with f =ω(X1, . . . , Xn) where Xk(p) = (p, ek), k = 1, . . . , n. Then we see that

[(ϕ∗ω) (X1, . . . , Xn)]p = [ω(dϕ(X1), . . . , dϕ(Xn))]p

= f ϕ(p)detÄ

ϕ′p(e1) · · · ϕ′p(en)ä

= f ϕ(p)detÄ

ϕ′pä

but also[(ϕ∗ω) (X1, . . . , Xn)]p = [ω(X1, . . . , Xn)]p = f (p)

for all p ∈ Rn, i.e.

f = f ϕ detÄ

ϕ′ä= f ϕ|det

Äϕ′ä|

where we used det(ϕ′) > 0 in the last equality. We finally yield, using thetransormation of coordinates theorem∫

Rnω =

∫Rn

f =∫

Rnf ϕ|det(ϕ′)| =

∫Rn

f =∫

Rnω

Definition 5.10 (Integral of ω ∈ Ωn0(M) with supp ω ⊂ U). If M is

oriented such that there is a chart (U, ϕ) that is consistent with the orentation,ω ∈ Ωn

0(M) and supp ω ⊂ U, then we define∫M

ω :=∫

Rn(ϕ−1)∗ω

Remark 5.11. The Integral is well defined, because assume (U, ϕ) is anotherchart such that γ = ϕ−1 and the change of coordinates ψ, then γ∗ω =(γ ψ)∗ω = ψ∗γ∗ω. By theorem 5.9 and since ψ is orientation preserving,we see that ∫

Rnγ∗ω =

∫Rn

γ∗ω

which makes the definition independent of the choice of (U, ϕ).

60


Definition 5.12 (Integral over ω ∈ Ωn0(M)). Let ω ∈ Ωn

0(M), (Uα, ϕα)α∈Ian open cover of M consisting of coodrinate charts and $1, . . . , $m a partitionof unity subordinate to (Uα)α∈I then we define

∫M

ω :=m∑

i=1

∫M

$iω

To ensure that our newly gained integral is well defined, we proof thefollowing theorem.

Theorem 5.13.∫

Mω thus defined is independet of the choices.

Proof. Without loss of generality we can assume (Uα)α∈I contains all coor-dinate neighbourhoods, i.e. the independence of (Uα)α∈I is no problem atall.For the independence of the partition of unity let $1, . . . , $m, $1, . . . , $m ∈C∞(M) be two partitions of unity, then

m∑i=1

∫M

$iω =m∑

i=1

∫M

Ñm∑

j=1$j

é$iω =

m∑i,j=1

∫M

$j$iω

=m∑

j=1

∫M

Ñm∑

i=1$i

é$jω =

m∑j=1

∫M

$jω

5.3 Stokes Theorem

61

5.3 Stokes Theorem

Definition 5.14 (Topological manifold with boundary). A topologicalmanifold with boundary is a second-countable, Hausdorff space which is lo-cally homeomorphic to an open subset of

Hn = (x1, . . . , xn) ∈ Rn | x1 ≤ 0 .

Definition 5.15 (General smoothness). Let A ⊆ Rn be any subset, thena map f : A → Rk is called smooth if there is an open subset U ⊆ Rn withA ⊆ U and a smooth map f : U → Rk with f |A = f .

Thus we know how to define a smooth manifold with a boundary.

Definition 5.16 (Smooth manifold with boundary). A smooth manifoldwith a boundary is a topological manifold with a boundary together with amaximal smooth atlas (Uα, ϕα)α∈I , Uα ⊂ M open, ϕα : Uα → ϕα(Uα) ⊂Hn

homeomorphic onto its image where coordinate changes are smooth.

Theorem 5.17 (Stoke’s theorem). Let M be an oriented manifold with bound-ary and ω ∈ Ωn−1

0 (M), then ∫M

dω =∫

∂Mω

Proof. As we habe the partition of unity as one of our tools, we may assumethat without loss of generality M = Hn = x ∈ Rn|x1 ≤ 0. For furthersimplification we also only consider the case n = 2 as the calculations workin the same manner for higher dimensions.We can write

ω = adx + bdy

62


thendω =

Ç∂b∂x− ∂a

∂y

ådx ∧ dy

By the definition of the integration of forms in Rn and Fubini’s theorem wenow see that ∫

H2dω =

∫ ∞

−∞

∫ 0

−∞

∂b∂x

dxdy−∫ 0

−∞

∫ ∞

−∞

∂a∂y

dydx

=∫ ∞

−∞b(0, y) dy− 0

=∫

∂H2b dy

=∫

∂H2ω

where we used the compact support of ω and the fact that the y-axis suitsthe induced orientation of ∂H2.

5.4 Fundamental theorem for flat vector bundles

Let E→ M be a vector bundle with connection ∇. Then

E trivial ⇐⇒ ∃ frame field Φ = (ϕ1, . . . , ϕk) with ∇ϕi = 0, i = 1, . . . , k

andE flat⇐⇒ E locally trivial.

Theorem 5.18 (Fundamental theorem for flat vector bundles). A vectorbundle

(E,∇) is flat ⇐⇒ R∇ = 0.

Proof.

”⇒”: Let (ϕ1, . . . , ϕk) be a local parallel frame field. Then we have fori = 1, . . . , k

R∇(X, Y)ϕi = ∇X∇Y ϕi −∇Y∇X ϕi −∇[X,Y]ϕi = 0.

Since R∇ is tensorial checking R∇ψ = 0 for the elements of a basis isenough.

”⇐”: Assume that R∇ = 0. Locally we find for each p ∈ M a neighborhoodU diffeomorphic to (−ε, ε)k and a frame field Φ = (ϕ1, . . . , ϕk) on U.Define ω ∈ Ω1(U, Rk×k) by

∇ϕi =k∑

j=1ϕjωji.

63


With ∇Φ = (∇ϕ1, . . . ,∇ϕk), we write

∇Φ = Φω.

Similarly, for a map F : U → Gl(k, R) define a new frame field:

Φ = ΦF−1

All frame fields on U come from such F. We want to choose F in sucha way that ∇Φ = 0. So,

0 != ∇Φ

= ∇(ΦF−1)

= (∇Φ)F−1 + Φd(F−1)

= (∇Φ)F−1 −ΦF−1dF F−1

= Φ(ω− F−1dF)F−1,

where we used that d(F−1) = −F−1dF F−1. Thus we have to solve

dF = Fω.

The Maurer-Cartan Lemma (below) states that such F : U → Gl(k, R)exists if and only if the integrability condition (or Maurer-Cartan equa-tion)

dω + ω ∧ω = 0

is satisfied. We need to check that in our case the integrability condi-tion holds: We have

0 = R∇(X, Y)Φ = ∇X∇YΦ−∇Y∇XΦ−∇[X,Y]Φ

= ∇X(Φω(Y))−∇Y(Φω(X))−Φω([X, Y])= Φω(X)ω(Y) + Φ(Xω(Y))−Φω(Y)ω(X)−Φ(Yω(X))−Φω([X, Y])= Φ(dω + ω ∧ω)(X, Y).

Thus dω + ω ∧ω = 0.

Lemma 5.19 (Maurer-Cartan). Let

U := (−ε, ε)n, ω ∈ Ω1(U, Rk×k), F0 ∈ Gl(k, R),

then

∃F : U → Gl(k, R) : dF = Fω, F(0, . . . , 0) = F0 ⇐⇒ dω + ω ∧ω = 0

64


Remark 5.20. Note that dω+ω∧ω automatically vanishes on 1-dimensionaldomains.

Proof. A general proof for arbitrary dimensions can be found in the differ-ential geometry II script. We will only proof the case for n = 2 here, thenthe situation is as follows:

F(0, 0) = F0,∂F∂x

= FA,∂F∂y

= FB

”⇒”: By the product rule we have

∂2F∂y∂x

=∂F∂y

A + F∂A∂y

= FBA + F∂A∂y

∂2F∂x∂y

=∂F∂x

B + F∂B∂x

= FAB + F∂B∂x

By Schwarz theorem we get that both equalities in fact are equal. Thiscan then be written as follows

∂B∂x− ∂A

∂y+ AB− BA = 0

because F is nonsingular.Compare this to

dω = d(Adx + Bdy) =Ç

∂B∂x− ∂A

∂y

ådx ∧ dy

ω ∧ω = (Adx + Bdy) ∧ (Adx + Bdy) = (AB− BA)dx ∧ dy

Thusdω + ω ∧ω = 0.

This last equation is also called the ”Maurer-Cartan equation” and issometimes also written as

∂A∂y− ∂B

∂x= AB− BA.

”⇐”: We use Picard-Lindelof to obtain a unique solution on the x-axis

G : (−ε, ε)→ Rk,k with G(0) = F0, G′(x) = A(x).

We use picard Lindelof again to obtain

F : (−ε, ε)2 → Rk,k with F(x, 0) = G(x).

65


For a fixed x this yields the linear ODE

∂F∂y

(x, y) = F(x, y)B(x, y).

We have to check that for all (x, y) ∈ (−ε, ε)2 it holds that

∂F∂x

(x, y) = F(x, y)A(x, y).

This is equivalent to showing that ∂F∂x − FA vanishes everywhere. By

construction we certainly have that

(∂F∂x− FA)(x, 0) = 0

for all x. But then also

∂

∂y(

∂F∂x− FA) =

∂2F∂x∂y

− ∂F∂y

A− F∂A∂y

=∂

∂y∂F∂x− ∂F

∂yA− F

∂A∂y

=∂

∂y(FB)− FBA− F

∂A∂y

=∂F∂y

B + F∂B∂y− FBA− F

∂A∂y

=∂F∂y

B + FÇ

∂B∂y− BA− ∂A

∂y

å=

∂F∂y

B− FAB

=

Ç∂F∂y− FA

åB

where we used that by Maurer-Cartan ∂B∂y − BA− ∂A

∂y = −AB. So wederived that

∂

∂y(

∂F∂x− FA) =

Ç∂F∂y− FA

åB

so for fixed x, the function

y 7→Ç

∂F∂x− FA

å(x, y)

satisfies the linear ODE above withÇ∂F∂x− FA

å(x, 0) = 0.

66


As the zero function is a solution, by the uniqueness of the solutionwe get

∂F∂x− FA = 0.

67

6. Riemann surfaces

Finally we are done with revising facts and theorems of differential geom-etry and start doing proper complex analysis.

Definition 6.1 (Riemann surface). A Riemann surface is a 2-dimensionalmanifold M with an almost complex structure J ∈ ΓEnd(TM).

A Riemann surface is not to be confused with the already known Rieman-nian surface that was defined as follows.

Definition 6.2 (Riemannian surface). A Riemannian surface is a 2-dimensionalmanifold with a Riemannian metric 〈., .〉.

Theorem 6.3.

a) A Riemann surface is oriented: There is a unique orientation such that forall X ∈ TM, X 6= 0, X, JX is positively oriented.

b) On a Riemann surface there is a unique conformal structure such thatfor each X ∈ TM the vectors X, JX are orthogonal and have the samelength (with respect to any given metric compatible with J). Equivalently:J = −J∗.

c) Given an oriented Riemannian surface, i.e. a Riemannian manifold ofdimension 2, then there is a unique almost complex structure J on M suchthat a) and b) apply.

Remark 6.4. Note that for any endomorphism

〈AX, Y〉 = 〈X, A∗Y〉

holds, thus (.)∗ does not depend on the conformal factor.

68

Riemann surfaces

By the theorem above we see that for a 2-dimensional manifold being aRiemann surface is really the same thing as having a conformal structureand an orientation.

Proof. Choose some Riemannian metric 〈., .〉∼ on M and define

〈X, Y〉 := 12(〈X, Y〉∼ + 〈JX, JY〉∼).

Then〈JX, Y〉 = 1

2(〈JX, Y〉∼ − 〈X, JY〉∼) = −〈JY, X〉

and thus J∗ = −J. Now define, σ ∈ Ω2(M), by

σ(X, Y) = 〈JX, Y〉

then this defines a volume form and hence an orientation on M. Further

σ(X, JX) = 〈JX, JX〉 = |X|2 > 0

menaing that X, JX are positively oriented. The uniqueness-part is left asan exercise. This proves a) and b). For c) we choose J to be the 90-degree-rotation in a positive sense.

Example 6.5. Let f : M → R3 be an oriented immersed surface then itsGauss map N defines a complex structure J by

d f (JX) := N × d f (X).

The example above raises the question if the almost complex structure weget is also complex.

Remark 6.6. Let V with 〈., .〉, dim V = 2, J ∈ End(V), J2 = −1, J∗ = −J.Then, if B ∈ End(V), B∗ = −B, we have B = λJ for some λ ∈ R.

Theorem 6.7. Let M be an oriented Riemannian surface. Then M is Kahler, i.e.∇J = 0 for the Levi-Civita connection ∇ of M.

69

Proof. Let X ∈ Γ(TM) and A ∈ ΓEnd(TM). Then

(∇A)∗ = ∇A∗

i.e. (.)∗ is parallel. This is because from 〈AX, Y〉 = 〈X, A∗Y〉 we get

〈∇(AX), Y〉+ 〈AX,∇Y〉 = 〈∇X, A∗Y〉+ 〈X,∇(A∗Y)〉

by differeniating both sides of the equation.Let B = ∇X J. Then

B∗ = (∇X J)∗ = ∇X J∗ = −∇X J = −B.

Thus B = λJ for some function λ. Moreover, J2 = −1 yields

0 = J∇J +∇J J = JB + BJ = 2λJ2 = −2λ .

Thus ∇J = B = 0.

Corollary 6.8. The Nijenhuis tensor of a Riemannian surface vanishes.

Topologically, the list of compact Riemann surfaces is as follows:

We will later proof the uniformization theorem which implies that the fol-lowing two surfaces are isometric.

Interestingly enough this does not generalize to surfaces of higher genus,for example the following two are not isometric.

70

Riemann surfaces

This is also true for a fat and a thin torus.

Given J and a volume form σ such that X, JX is positively oriented. Then

〈X, Y〉 := σ(X, JY)

defines a Riemannian metric. To see this consider a 2-dimensional realvector space V with determinant det. Let A ∈ End(V). Then

det(X, AY)− det(Y, AX) = trA det(X, Y)

For an almost complex structure J Cayley-Hamilton yields

0 = J2 − tr(J)J + det(J)I = (det(J)− 1)I − tr(J)J.

Thusdet(J) = 1 and tr(J) = 0.

as I and J are linearly independent. In particular J is skew with respect todet. Same holds for σ and J above. Thus 〈., .〉 is symmetric. Moreover,

〈X, X〉 = σ(X, JX) > 0.

We make this a theorem.

Theorem 6.9. Let (M, J) be a Riemann surface and σ a volume form on M,such that σ(X, JY) > 0 for X 6= 0, then

〈X, Y〉 = σ(X, JY)

defines a Riemannian metric in the conformal class of (M, J).

6.1 Holomorphic line bundles over a Riemann sur-face

Definition 6.10 (Complex line bundle). A Complex line bundle over a man-ifold M is a rank-2 vector bundle L with complex structure J ∈ ΓEnd(L).

Remark 6.11. By definition, in a complex line bundle all fibers are 1-dimensionalcomplex vector spaces, i.e. complex ”lines”.

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6.1 Holomorphic line bundles over a Riemann surface

Example 6.12. We gather some examples of complex line bundles over aRiemann surface:

(i) L = M×C, then Γ(L) = C∞(M; C).

(ii) TM itself is a complex line bundle.

(iii) T∗M, then Γ(T∗M) = Ω1(M; C).

Definition 6.13 (complex connection on a complex vector bundle). LetE be a complex vector bundle. A connection ∇ on a complex vector bundle iscalled complex is for all ψ ∈ ΓE we have

∇(Jψ) = J∇ψ

or in other words ∇J = 0.

Proposition 6.14. Locally every complex vector bundle is isomorphic to U×Ck

Proof. Choose a real frame field φ1, . . . , φ2k, without loss of generality wecan assume that φ1,p, Jφ1,p, . . . , φk,p, Jφk,p is a basis of Ep. Then, by con-tinuity, this will hold on a neighborhood of p, which yields a complexframe.

Definition 6.15 (complex trivial). A complex vector bundle (with connec-tion) is complex trivial if it is isomorphic as a complex vector bundle (withconnection) to M×Ck.

Consider two complex connections ∇ and ∇ on (E, J), then we alreadyknow that ∇ = ∇ + A, meaning that they differ by an End(E)-valued 1-form

A = ∇ −∇ ∈ Ω1(E, End(E)).

We observe that

J∇ψ + AJψ = ∇(Jψ) + A(Jψ) = ∇(Jψ)

= J∇ψ = J(∇ψ + Aψ) = J∇ψ + JAψ

thusAJψ = JAψ

meaning that in fact even

A ∈ Ω1(E, End+(E)).

Conversely, if ∇ is a complex connection and A ∈ Ω1(E, End(E)), then∇+ A is a complex connection.

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Riemann surfaces

Theorem 6.16. Any complex vector bundle has a complex connection.

Proof. Use the local triviality and a partition of unity.

Definition 6.17 (Curvature tensor). The curvature tensor R∇ of a connec-tion on a vector bundle E over a manifold M is defined as the map

R∇ : Γ(TM)× Γ(TM)× Γ(E)→ Γ(E),

(X, Y, ψ) 7→ ∇X∇Yψ−∇Y∇Xψ−∇[X,Y]ψ

The curvature tensor R∇ of a connection ∇ on a vector bundle can beviewed as an End(E)-valued 2-form:

R∇ ∈ Ω2(M; End(E)) .

Proposition 6.18. Let E be a complex vector bundle and ∇ be a complex con-nection on E. Then if ∇ commutes with J, so does R∇, i.e.

∇J = J∇ ⇒ R∇ ∈ Ω2(M; End+(E)).

Proof. For X, Y ∈ Γ(TM) and ψ ∈ Γ(E) it is

R(X, Y)(Jψ) = ∇X∇Y(Jψ)−∇Y∇X(Jψ)−∇[X,Y](Jψ) = JR(X, Y)ψ.

6.2 Poincare-Hopf index theorem

Throughout this section M denotes a compact oriented 2-dimensional man-ifold (without J) and L denotes a rank 2 vector bundle over M with J, i.e.a complex line bundle. The endomorphism bundle End(L) splits into com-plex linear End+(L) and complex anti-linear part End−(L),

End(L) = End+(L)⊕ End−(L), End±(L) = A ∈ End(L) | AJ = ±JA .

In particular, End+(L)p = spanIp, Jp and Lp is a complex vector spacesetting

(α + iβ)ψ := (αI + βJ)ψ .

So, for a complex line bundle L we have

End+(L) ∼= M×C

and therefore for a complex connection ∇ on L there is Ω ∈ Ω2(M; C) suchthat for all X, Y ∈ Γ(TM), ψ ∈ Γ(L)

R∇(X, Y)ψ = −Ω(X, Y)ψ .

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Definition 6.19 (Curvature 2-form). The unique Ω ∈ Ω2(M, E) such thatfor all X, Y ∈ Γ(TM), ψ ∈ Γ(L) R∇(X, Y)ψ = −Ω(X, Y)ψ holds is calledthe curvature 2-form of R∇.

If dim M = 2 and M is oriented we can compute∫M

Ω ∈ C .

Interestingly this number is independent of the particular choice of theconnection, although Ω depends on R∇. We can see this by the followingcomputation:Let ∇ = ∇+ α for α ∈ Ω(M, C). then we get

R∇(X, Y)ψ = ∇X(∇Yψ)− ∇Y(∇Xψ)−∇[X,Y] − α([X, Y])ψ

= R∇(X, Y)ψ +∇(α(Y)ψ) + α(X)α(Y)ψ + α(X)∇Yψ

−∇Y(α(X)ψ)− α(Y)α(X)ψ− α([X, Y])ψ

= R∇(X, Y)ψ + (Xα(Y)−Yα(X))ψ− α([X, Y])ψ

= R∇(X, Y)ψ + dα(X, Y)ψ

So if ∇ = ∇+ α for α ∈ Ω1(M; C), then R∇ = R∇ + dα. Hence

Ω = Ω− dα

hence by Stokes’ theorem ∫M

Ω =∫

MΩ.

Definition 6.20 (degree of a complex line bundle). If L is a complex linebundle over an oriented compact surface, then

deg(L) := 12πi

∫M

Ω

is called the degree of L.

Remark 6.21. The Poincare-Hopf index theorem will tell us that deg(L) isactually an integer. Bur for now, the intermediate goal will be to show thatit is real.

Choose a euclidean fiber metric 〈., .〉 on L that is compatible with J. Weknow that there is one such and that all other possible choices are of theform 〈., .〉∼ = e2u〈., .〉. Given a complex connection ∇, then we define

(∇X〈., .〉)(ϕ, ψ) := X 〈ϕ, ψ〉 − 〈∇X ϕ, ψ〉 − 〈ϕ,∇Xψ〉 .

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Riemann surfaces

This is not yet positive definite, but already (∇X〈., .〉) is symmetric, bilinearand invariant under J. So ∇X〈., .〉 has all properties os 〈., .〉 except for thatis not necessarily positive definite. Consequently we find an ω ∈ Ω1(M, R)such that

∇〈., .〉 = ω〈., .〉.What happens to ∇〈., .〉 of we change ∇ to ∇ = ∇+ α? By the definitionwe have, using α = β + iγ, that

(∇X〈., .〉)(ϕ, ψ) = (∇X〈., .〉)(ϕ, ψ)− 〈α(ψ), ϕ〉 − 〈ψ, α(X)ϕ〉= (∇X〈., .〉)(ψ, ϕ)− 2β(X) 〈ψ, ϕ〉

Thus if we choose β(X) = 12 ω we get that ∇ is a metric connection in L. So

without loss of generality we can assume that ∇ is compatible with J and〈., .〉. As deg does not depend on ∇ we can just choose it like this.

So as a quick summary, consider a complex line bundle L with 〈., .〉 com-patible with J and a complex connection ∇ that is metric with respect to〈., .〉. Then we want to show that

R∇ = −Ω with Re(Ω) = 0.

This means that the curvature 2-form of R∇ is purely imaginary.

Theorem 6.22. If, on a complex line bundle L, ∇ is complatible with J and〈., .〉, then Ω is purely imaginary.

Proof. By a straightforward calculation we yield

〈R(X, Y)ψ, ϕ〉 =¨∇X∇Yψ−∇Y∇Xψ−∇[X,Y]ψ, ϕ

∂= X 〈∇Yψ, ϕ〉 − 〈∇Yψ,∇X ϕ〉 −Y 〈∇Xψ, ϕ〉〈∇Xψ,∇Y ϕ〉 − [X, Y] 〈ψ, ϕ〉+

¨ψ,∇[X,Y]ϕ

∂= XY 〈ψ, ϕ〉 − X 〈ψ,∇Y ϕ〉 −Y 〈ψ,∇X ϕ〉+ 〈ψ,∇Y∇X ϕ〉− yX 〈ψ, ϕ〉+ Y 〈ψ,∇x ϕ〉+ X 〈ψ,∇Y ϕ〉 − 〈ψ,∇X∇Y ϕ〉− [X, Y] 〈ψ, ϕ〉+

¨ψ,∇[X,Y]ϕ

∂= − 〈ψ, R(X, Y)ϕ〉

In fact, for general vector bundles with metric and metric connection ∇ itholds that for R(X, Y) ∈ Γ(E) it is

R(X, Y)∗ = −R(X, Y)

, i.e. R takes values in the skew-adjoint endomorphisms. In our case, withΩ = α + Jβ ∈ C we get

R(X, Y)ψ = −Ω(X, Y)ψ = (α(X, Y) + β(X, Y)J)ψ

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soR(X, Y)∗ = −R(X, Y) ⇔ α = 0.

Thus Ω is purely imaginary and we get the following corollary.

Corollary 6.23.deg(L) = 1

2πi

∫M

Ω ∈ R .

The original claim was that L is even an integer and to proof this we aregoing to ”count” it. But before, we will ”steal” the following result fromdifferential topology, that are hard to proof, but easy to believe. So we willsave the time and omit the proofs.

Lemma 6.24 (Sard). Let U ⊂ Rn open, if f : U → Rk ∈ C∞, then the sety ∈ Rk | ∃x ∈ U with f (x) = y and rank f ′(x) < k

of critical values of f

has measure zero.

With some work one can use Sard’s lemma to proof the following theorem.

Theorem 6.25 (Transversality theorem). Call two submanifolds M1, M2 ⊂M transversal if for all p ∈ M1 ∩M2 we have TpM1 + TpM2 = TpM. GivenM1, M2 ⊂ M one can perturb M1 slightly such (refering to C∞-topology) suchthat afterwards M1 and M2 are transversal.

If we apply the transversality theorem toM = total space of L, dim M = 4M1 =

¶0p | p ∈ M

©graph of the zero section

M2 =¶

ψp midp ∈ M©

graph of some section ψ ∈ Γ(L)

Then we now there is some section ψ transversal to the zero section. Thismeans that locally M = U ⊂ R2 (including orientation), L = U ×C then ψcan be viewed as a map

ψ : U → C

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Riemann surfaces

In this case we have that M1 is transversal to M2 if and only if det ψ′(p) 6= 0.

Definition 6.26 (Winding number). We define the winding number of asection ψ ∈ Γ(L) that is transversal to the zero section as

indpψ :=

1, det ψ′(p) > 0−1, det ψ′(p) < 0

.

Remark 6.27. The winding number indpψ is independetn of oriented charts.

More generally, if p is an isolated zero of ψ (seen as a map ψ : U → C)define

indpψ =1

2πi

∫|z−p|=$

dψ

ψ

which is known from the complex analysis 1 lecture.

Theorem 6.28 (Poincare-Hopf index theorem). Let L be a cmplex line bundleover a compact oriented surface and ∇ a complex connection with curvature 2-form Ω ∈ Ω2(M, C) and ψ ∈ Γ(L) with isolated zeroes p1, . . . , pn. Then

12πi

∫M

Ω =n∑

i=1indpi ψ ∈ Z.

Remark 6.29. Note that by the compactness of the surface ψ can only havefinitely many isolated zeroes.

Remark 6.30 (Contemplate consequences).

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i) deg(L) ∈ Z and depends only on L as a oriented rank2 real vectorbundle, i.e. is independent of J and ∇. (A change of the orientationchanges the sign of deg(L))

ii) From the fact that it does not depend on ψ, counted with multiplicity,all sections with only finitely many zeros ψ ∈ Γ(L) have the samenumber of zeroes.

Definition 6.31 (Euler characteristic). Given an oriented compact surfaceM, define the Euler characteristic χ(M) as

χ(M) := deg(TM).

Remark 6.32. If we change the orientation of M, also the orientation of TpMis changed, therefore indpX is unchanged. Hence χ(M) is independent ofthe orientation.

Theorem 6.33 (Gauß-Bonnet). If M is an oriented compact Riemannian sur-face with volume form det and curvature form Ω = iK det of the Levi-CivitaConnection ∇, for the Gaussian curvature K ∈ C∞(M), then∫

MK det = 2πχ(M).

Example 6.34. Consider M = S2, then the Gaussian curvature K ≡ 1 and

area(S2) =∫

S21 det = 4π = 2π · 2

as χ(S2) = 2.

Remark 6.35. The setup for the Poincare-Hopf index theorem is the fol-lowing: M is a compact, oriented 2-dimensional manifold (without anycomplex structure J), L is a complex line bundle, i.e. rank2-vector bundle

78

Riemann surfaces

over M with almost complex structure J.Now we make choicesChoose a connection ∇ on L with ∇L = 0

Choose a connection ψ ∈ Γ(L) with isolated singularities p1, . . . , pn ∈ M

By Poincare-Hopf, for the volume 2-form Ω ∈ Ω2(M, C) of the curvaturetensor R∇ we have

12πi

∫M

Ω =n∑

j=1indpj ψ =: deg(L) ∈ Z

We notice that the left hand side of the equation does not depend on thechoice of ψ and the middle part does not depend on the connection ∇,hence the right hand side deg does neither depend on the choice of ψ, noron the choice of ∇.

Proof. We choose open neighborhoods U1, . . . , Un ⊂ M of p1, . . . , pn respec-tively such that

(i) there is ϕ ∈ Γ(L|Uj) without zeros, i.e. there exists a 1-dimensionalframe field on Uj

(ii) there is a chart φj : Uj → R2 such that φ(pj) = 0 andBε(0) =

¶(x, y) ∈ R2 | x2 + y2 ≤ ε2© ⊂ φj(Uj) for 0 < ε < 1.

(iii) Uj ∩Uk = ∅ for j 6= k

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We define

Mε := M \

n⋃j=1

φ−1j (Bε(0))

and see that this defines a compact manifold with boundary. Further ψ|Mε

has no zeros.Define a 1-form η ∈ Ω1(Mε, C) by

∇ψ = ηψ.

For X, Y ∈ Γ(TM) we compute

−Ω(X, Y)ψ = R∇(X, Y)ψ= ∇X∇Yψ−∇Y∇Xψ−∇[X,Y]ψ

= ∇Xη(Y)ψ−∇Yη(X)ψ− η([X, Y])ψ= (Xη(Y)−Yη(X)− η([X, Y]))ψ + η(Y)η(X)ψ− η(X)η(Y)ψ= dη(X, Y)ψ

where we used that η(Y)η(X)ψ − η(X)η(Y)ψ = 0 as η takes values in C

and the multiplication on C is commutative. So we have that

Ω = −dη.

This immediately calls for stokes theorem! First we define Bεj := φ−1

j (Bε(0)),then ∫

Mε

Ω = −∫

Mε

dη = −∫

∂Mε

η =n∑

j=1

∫∂Bε

j

η

Note that for the last equation, the orientation yields a sign-flip.

For fixed ε > 0 and j we define

ψ|∂Bεj= γϕj.

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Riemann surfaces

Further we assume that ∂Bεj = S1. We use the fact that we can compute

the winding number using the integral that counts zeroes and see that withstokes theorem

indpj ψ =1

2πi

∫S1

dγ

γ

=1

2πi

∫S1(d log |γ|+ idarg(γ))

=1

2π

∫S1

darg(γ)

For X ∈ Tq∂Bεj ⊂ TM we have

∇Xψq = dγ(X)ϕj(q) + γ(q)∇X ϕj(q) = dγ(X) + ϕj(q) + γ(q)ω(X)ϕj(q)

where ω ∈ Ω1(Uj) is 1-form defined by

ω(X)ϕj(q) = ∇X ϕj(q).

On the other hand

∇Xψq = η(X)ψq = η(X)γ(q)ϕj(q).

As ϕj(q) appears everywhere we can cancel it and derive

γη = dγ + γω ⇔ η =dγ

γ+ ω.

That is fortunate, because then∫∂Bε

j

η =∫

∂Bεj

dγ

γ+∫

∂Bεj

ω = indpj ψ2πi +∫

∂Bεj

ω.

So it remains to show that

limε→0

∫∂Bε

j

ω = 0.

This is again an application of stokes. As ω is also smooth on the interiorof Uj we have ∫

∂Bεj

ω =∫

Bεj

dω → 0

for ε→ 0.

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For now consider a 2-dimensional manifold M and equip the tangent bun-dle TM with a J. This turns M into a Riemann surface. We now will applythe Poincare-Hopf index theorem to L = TM.

Remark 6.36. Recall that the Euler characteristic χ(M) is the index sum ofany vector field with only isolated zeroes. In particular, χ(M) is indepen-dent of J.

Another consequence of the transversality theorem is that on every man-ifold there is a height function h ∈ C∞(M) such that dh vanishes only atisolated points. For the case dim M = 2 consider an immersion f : M→ R3

with normal field N : M → S2. We can manufacture a height function h asabove as follows:Choose some regular value a ∈ S2 of N (i.e. N(p) = a⇒ dpN has full rank)and define

h := 〈 f , a〉 .

In the sketch we have 8 critical points of gradh, that is when gradh is verti-cal, hence

indp gradh =

1 if h has a min or max at p.−1 if h has a saddle point at p.

Example 6.37. Consider the function h(x, y) = x2 − y2. This h has a saddleat (0, 0). The level sets look more or less like this

Corollary 6.38. The Euler characteristic of a Riemann surface can be expressedas

χ(M) = |min|+ |max| − |saddles|.

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Riemann surfaces

Definition 6.39 (k-cell).

i) A 2-cell F in a surface M is a subset diffeomorphic to a convex polygonin R2.

ii) A 1-cell e in a surface M is a subset diffeomorphic to [0, 1].

iii) A 0-cell p in a surface M is a point in M.

Definition 6.40 (Cell-decomposition). A cell-decomposition of a complexsurface with boundary M is a collection C1, . . . , Cm of 0−, 1−, or 2-cells suchthat

m⋃i=1

Ci = M

and for each i, j ∈ 1, . . . , m with i 6= j there is k ∈ 1, . . . , m such thatCi ∩ Cj = Ck and dim Ck < max

¶dim Ci, dim Cj

©.

Definition 6.41 (triangulation). A triangulation of a complex surface M iscell-decompostion of M in cells of dimesnion at most 2.

Given a triangulation of a compact oriented surface with F being the setof 2-cells (”faces”), E the set of 1-cells (”edges”) and V the set of 0-cells(”vertices”) one can construct X ∈ Γ(TM) with

a source at each face centera sink at each vertexa saddle at each edge center

83


This yields the more popular formula for the Euler characteristic

χ(M) = V − E + F.

84

7. Classification of Line Bundles

We want to classify line bundles up to isomorphisms. The final result willbe that deg is the only invariant possible, hence L ∼= L′ if and only ifdeg L = deg L′. Further we show that for all z ∈ Z there is a bundle Lzsuch that deg Lz = z.But first it will be necessary to briefly revise some topics of (multi-)linearalgebra.

7.1 Tensor products of vector spaces and bundles

Definition 7.1 (tensor product of vector spaces). Let V, W be finite di-mensional K-vector spaces with K ∈ R, C. Then

V ⊗W := β : V∗ ×W∗ → K | β bilinear

This leads to the infamous bra-ket notation known from e.g. quantummechanics. For α ∈ V∗ and v ∈ V we define

α(v) := 〈α|v〉 .

From this, to emphasize that α is a dual-vector we can also write 〈α|, anal-ogously for a vector v we write |v〉. See also the correspondence to theearlier used musical isomorphisms [ and ]. In total we have the followingidentity

〈u, v〉 =⟨

u[∣∣∣v⟩ = u[(v) = u[v.

Definition 7.2 (tensor product). Let v ∈ V and w ∈ W, then for αinV∗,β ∈W∗ we define v⊗ w ∈ V ⊗W by

v⊗ w(α, β) := α(v) · β(w) = 〈α|v〉〈β|w〉 .

85

7.1 Tensor products of vector spaces and bundles

If a1, . . . , an is a basis of V and b1, . . . , bm is a basis of W then

ai ⊗ bj, 1 ≤ i ≤ n, 1 ≤ j ≤ m

is a basis of V ⊗W which implies that dim V ⊗W = dim V · dim W.

Remark 7.3. The tensor product of line bundles yields again a line bundle.

Let for now L1 and L2 be two line bundles over a manifold M, then L1⊗ L2is a line bundle over M as well.

Proposition 7.4. If L1 is isomorphic to L1 and L2 is isomorphic to L2 thenL1 ⊗ L2 is isomorphic to L1 ⊗ L2.

Proof. The proof should be clear.

So the tensor product is well defined on isomorphism-classes of line bun-dles over M.

Now consider the trivial bundle M×C, then there is an isomorphism be-tween

(M×C)× L→ L.

This can be seen as follows:Let βin(M×C)p × Lp, then for ω ∈ C∗ and η ∈ L∗p it is β(ω, η) ∈ C and

f (β) = β(ω, ·) ∈ (L∗p)∗ = Lp.

as f (β) only has an input for elements of the second slot (Note that ω(1) =1 hence ”ω = 1∗”). So (M×C)× L is canonically isomorphic to L itself.Similarly, for β ∈ (L⊗ L)p we construct

f (β) ∈ (L⊗ L)p

by( f (β))(ω, ω) := β(ω, ω).

So switching arguments yields an isomorphsim.

To summarize this: The isomorphism classes of complex (or real) line bun-dles over a connected manifold M form an abelian group under the tensorproduct. The trivial bundle (or any bundle isomorphic to it) is the neutralelement and L∗ =: L−1 is the inverse element of L.

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Classification of Line Bundles

7.2 Line Bundles on Surfaces

The intermediate goal will be to proof that the map deg is an isomorphismof abelian groups between the set of equivalence classes of line bundleswith ⊗ and (Z,+). Some preparation is needed beforehand.

Lemma 7.5. If M is a connected manifold and U ⊂ M open, p ∈ M. Thenthere is a diffeomorphism f : M→ M such that f (U) ⊂ U and f (p) ∈ U.

Proof. Since M is connected, there is a smooth γ : [0, 1]→ M with γ(0) = pand γ(1) =: q ∈ U. By the transversality theorem, γ can be chosen as animmersion with transversal self-intersections. Resolve the self-intersectionsto get a collection of embedded curves and delete loops to obtain an em-bedded γ.

By the ”tubular neighborhood theorem” we find a diffeomorphism

g : [−ε, 1 + ε]× Dn−1 → V ⊂ M with

g(0, 0) = pg(1, 0) = q

.

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Now construct a diffeomorphism

f : [−ε, 1 + ε]× Dn−1 → [−ε, 1 + ε]× Dn−1

such that f (x) = x, for x in some neighborhood of ∂Ä[−ε, 1 + ε]× Dn−1ä

f (0, 0) = (1, 0)

Then define

f (r) :=

g f g−1(r) for r ∈ Vr else

then f is smooth and does the trick.

Proposition 7.6. If M is a smooth and connected n-dimesnional manifold andp1, . . . , pm ∈ M, then there is a U ⊂ M diffeomorphic to Dn such thatp1, . . . , pm ∈ U.

Proof. We will proof this by induction on m ∈ N. For m = 1 the statementis clear. By the induction hypothesis p1, . . . , pm−1 are already contained insome U. Use the above Lemma to find a diffeomorphism f : M → M withU := f (U) ⊂ U and f (pm) ∈ U. Then p1, . . . , pm ∈ U, hence the claim isproven.

Theorem 7.7. Let E be a vector bundle over the closed n-dimensional unit discDn, then E is trivial.

Proof. Choose a basis φ1, . . . , φk of E0 and any connection ∇ on E. Thenthere are unique sections ϕ1, . . . , ϕk ∈ Γ(E) such that ϕj(0) = φj such that∇X ϕj = 0 when Xp = p is the position vector field on Dn. Then ϕ1, . . . , ϕkform a frame field.

Now we are able to proof the desired theorem.

Theorem 7.8. The map [L] 7→ deg L ∈ Z is an isomorphism of abelian groups.

88


Proof. We will give two proofs for the homeomorphism property:

i) Let ψ ∈ Γ(L) and ψ ∈ Γ(L) with isolated zeros. Then we immediatelyget

indp(ψ⊗ ψ) = indp ψ + indp ψ.

This is because the lefthand side has a zero if either of the two vec-tor fields has a zero and a multiplication of winding numbers yields”winding number of γ · γ = winding number of γ + winding numberof γ”.

ii) The second proof uses curvature. Choose complex connections ∇ onL and ∇ on L. Then there is a unique connection ∇ on L ⊗ L suchthat for all ψ ∈ Γ(L), ψ ∈ Γ(L) we have

∇(ψ⊗ ψ) = (∇ψ)⊗ ψ + ψ⊗ (∇ψ).

Then

R∇(X, Y)(ψ⊗ ψ)

= ∇X∇Y(ψ⊗ ψ)− ∇Y∇X(ψ⊗ ψ)− ∇[X,Y](ψ⊗ ψ)

= ∇XÄ(∇Yψ)⊗ ψ + ψ⊗ (∇Yψ)

ä(ψ⊗ ψ)

− ∇YÄ(∇Xψ)⊗ ψ + ψ⊗ (∇Xψ)

ä(ψ⊗ ψ)− ∇[X,Y](ψ⊗ ψ)

= ∇X∇Yψ⊗ ψ +∇Yψ⊗ ∇Xψ +∇X ⊗ ∇Yψ + ∇X∇Yψ

−∇Y∇Xψ⊗ ψ−∇Xψ⊗ ∇Yψ−∇Y ⊗ ∇Xψ− ∇Y∇Xψ

− (∇[X,Y]ψ)⊗ ψ− ψ⊗ (∇[X,Y]ψ)

= R∇(X, Y)ψ⊗ ψ + ψ⊗ R∇(X, Y)ψ

which implies thatΩ = Ω + Ω

and yieldsdeg(L⊗ L) = deg L + deg L.

It remains to show that deg is surjective and injective. We will start withthe anterior.To show that deg is surjective we use the ”skyscraper construction” to con-struct a vector bundle of arbitrary degree d. Choose a point p ∈ M andtake (M \ p) × C, i.e. the trivial bundle over M \ p. also, for someopen neighborhood U of p take U ×C. Consider now

[(M \ p)×C]∐

[U ×C]

= (0, q, z) | q ∈ M \ p , z ∈ C ∪ (1, q, w) | q ∈ U, w ∈ C .

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We define an equivalence relation ∼ on it by

(0, q, z) ∼ (1, q, w) :⇔ q = q, w = ϕ(q)dz.

where vφ is a diffeomorphism ϕ : U → V ⊂ R2 with 0 ∈ V.

Note that q 6= p hence ϕ(q)d 6= 0. Let now

L := equivalence classes

then

π : L→ M,

π[(0, q, z)] := qπ[(1, q, w)] := q

is well defined, hence L is a line bundle. Now define ψ ∈ Γ(L) by

ψp =

[(0, q, 1)], forq 6= p[(1, q, ϕ(q)d)] forq ∈ U

then ψ has a zero of degree d at p, hence L has deg L = d > 0 and deg L−1 =−d which yields the surjectivity.It remains to show that deg is injective and since it is already clear that it isa homeomorphism of groups, it suffices to show the following statement:

deg L = 0 ⇔ L has a nowhere vanishing section ψ ∈ Γ(L).

By the transversality theorem we can choose a section ψ ∈ Γ(L) with iso-lated zeros p1, . . . , pn. Then by assumption

indp1 ψ + . . . + indpn ψ = 0.

By the preparation we did beforehand we can apply a diffeomorphismM → M such that p1, . . . , pn ∈ U for some U diffeomorphic to Dn. Inparticular this can also be done such that this is true for some U1 ⊂ U.

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Because U is diffeomorphic to Dn we can choose some section ϕ ∈ Γ(L|U)without zeros, as L|U is trivial. Then there is g ∈ C∞(U, C) such that

ψ|U = g · ϕ.

As the total number of zeros is zero, just as in the proof of the Poincare-Hopf index theorem we get that the winding number

0 =∫

∂U1

dgg

which means that ∫γ

dgg

= 0

for all closed curves γ in U \U1. S0 there exists f ∈ C∞(u \U1, C) with

g = e f .

Let now σ ∈ C∞(R) with σ(x) = 0 for x < 2 and σ(x) = 1 for x > 3. Define

f : U → C by f (p) = σ(|z(p)|) f (p).

Then a smooth section ψ ∈ Γ(L) of L can be obtained by setting

ψ(p) =

e f (p)ϕp for p ∈ Uψp else

Further ψ has no zeros, hence is a frame.

7.3 Combinatorial Topology

The setup for the first part of this section will be a compact oriented smoothsurface M. We have already learned what a cell-decomposition of such asurface is. In order to proof the existence of one such we need to work offthe following steps:

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i) Choose a Riemannian metric. As M is compact, this gives a distancefucntion d : M×M→ R.

ii) Choose p1, . . . , pn ∈ M such that d(pi, pj) < ε for all i, j and a given εwhich is sufficiently small.

iii) Define so called Voronoi-cells by

Φj :=¶

q ∈ M | d(pj, q) ≤ d(pi, q) for all i ∈ 1, . . . , n©

.

These cells form the faces of the cell-decomposition.

The points p1, . . . , pn are the vertices of the dual cell-decomposition. Verticesof the original cell-decomposition correspond 1− 1 to faces of the dual cell-decomposition and vice versa.

Given a cell-decomposition of M, define E to be the set of oriented edges.

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This gives a bijective map$ : E→ E

without fixed points that assigns to each edge ”the same” edge with differ-ent orientation. It is easy to see that

$2 = $ $ = IdE

hence $ is an involution. Additionally we have a permutation

σ : E→ E

which is defined as follows:find the face ϕ which is to the left of edefine σ(e) as the oriented edge following e in the edge-cycles of ϕ.

So in summary, a cell-decomposition of a compact oriented surface M gives:

i) finite set E

ii) $ a fixed point free involution of E

iii) σ a permutation of E

Remark 7.9. For every such tripleÄE, $, σ

äthere is a compact oriented sur-

face with a cell-decomposition (includes di- or one-gons) which is uniqueup to a diffeomorphism.

We have the following 1− 1 correspondences:

F := faces of cell-decomposition ↔ cycles of σ

E := edges of cell-decomposition ↔ cycles of $

Example 7.10. Consider E = e+, e−, then $(e+) = e− and $(e−) = e+.Then there are two possible cases of how this surface looks like:

i) σ = IdE:

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ii) σ = $:

To avoid such degenerate examples one can ask for∣∣∣E∣∣∣ ≥ 3.

We see that the composition$ σ

rotates an edge e clockwise around its starting vertex. Thus we obtain theset

V := vertices of cell-decomposition ↔ cycles of σ $

In particular we know the cardinality of V, E and F which leads to thefollowing theorem.

Theorem 7.11. For a compact oriented surface, the Euler-characteristic χ =|V| − |E|+ |F| is even.

Proof. We know that sgn $ = (−1)|E|. If now σ is a cycle of length k, thensgn σ = (−1)k+1. The face ϕj has k j edges. Then

sgn σ = (−1)(k1+1)+...+(k|F|+1) = (−1)|E|+|F| = (−1)|F|

where the last equality inherits from the fact that E must be even. Similarlywe get

sgn σ $ = (−1)|V|.

As sgn is a homeomorphism from Sn → Z/2Z we get

(−1)|V| = sgn σ $ = sgn σsgn $ = (−1)|F|(−1)|E|

which means that

|V| = |F|+ |E| = −|F|+ |E| mod 2

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Let us for now restrict our attention to a solely discrete oriented surface

M = (E, $, σ)

with a finite set E, bijective maps $, σ : E → E whereby $ has no fixedpoints. We obtain the sets

E = cycles of $ ,F = cycles of σ ,V = cycles of $ σ .

Definition 7.12 (dual surface). The dual discrete surface M∗ is given by

M∗ =ÄE, $ =: $∗, $ σ =: σ∗

ä.

Definition 7.13 (subsurface). A finite set E ⊆ E is called a subsurface of Mof $(E) = E and σ(E) = E.

Remark 7.14. Then the M =ÄE, $|E, σ|E

äis also a discrete surface.

Definition 7.15 (connected discrete surface). A discrete surface M is calledconnected if it has no proper subsurfaces.

Remark 7.16. Every discrete surface is the disjoint union of its connectedcomponents.

Theorem 7.17. A discrete surface M is connected if and only if its dual surfaceM∗ is connected.

Proof. The invariance of a subsurface E of M under $ and σ gives

σ∗(E) = $ σ(E) = $(E) = E

as well as$∗(E) = $(E) = E.

This conclusion is symmetric in M and M∗ which yields the claim.

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7.4 Discrete Forms

7.4 Discrete Forms

Having discrete surfaces defined it is only natural to aim for a definition ofdiscrete forms as in the smooth case these are the natural choice to integrateover a surface. For a discrete surface M just like in the previous section wedefine

Ω0(M) := f : V → RΩ1(M) :=

¶ω : E→ R

©Ω2(M) := η : F → R

But given these definitions, what does it mean to have a discrete form?When M comes from a cell decomposition, then a discrete 2-form η is justlike a smooth 2-form where the only information we keep is

∫ϕ

η

for each fache ϕ ∈ F. Hence, for each face ϕ ∈ F there is a real number R.For a discrete 1-form ω this can be motivated by only keeping the informa-tion ∫

eω.

Here, as we find many forms that agree on e with both orientations, discrete1-forms correspond to equivalence classes of smooth 1-forms that agree onall edges in both possible orientations.

In the smooth setup, on a connected, oriented surface M an ω ∈ Ω1(M) isexact if and only if for every closed curve γ : [0, 1] → M with γ(0) = γ(1)we have ∫

γω =

∫ 1

0ωγ(t)γ

′(t)dt = 0.

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We aim for a discrete analogue of this well-known theorem. For an orientededge e ∈ E we definestart(e) := orbit of $(e) under $ σ

end(e) := orbit if e under $ σ

Definition 7.18 (discrete path).

i) A discrete path in M is a sequence γ = (e1, . . . , en) with ei ∈ E suchthat end(ei) = start(ei+1) for i = 1, . . . , n− 1.

ii) A path γ is called closed if end(en) = start(e1).

Remark 7.19. Note that a path γ is a sequence of edges, not vertices!

Definition 7.20 (discrete exterior derivative d0). For a 0-form f ∈ Ω0(M)define d f ∈ Ω1(M) by

d f (e) := f (end(e))− f (start(e)) .

Theorem 7.21. M is connected if and only if for all pairs of points p, q ∈ Vthere is a path γ in M with start(γ) = p and end(γ) = q.

Proof. The proof will be left as an exercise.

Definition 7.22 (integral of a discrete 1-form). Let γ = (e1, . . . , en) be apath in M and ω ∈ Ω1(M), then

∫γ

ω =n∑

i=1ω(ei).

Now we can proof the desired analogue of the smooth case.

Theorem 7.23. On a connected M we have that a discrete 1-form ω ∈ Ω1(M)is exact if an only if ∫

γω = 0

for all closed paths γ in M.

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7.5 Poincare Duality

Proof.

”⇒”: Let ω be exact, i.e. ω = d f for some f ∈ Ω0(M) and γ a closed pathin M. Then∫

γω =

n∑i=1

ω(ei) =n∑

i=1d f (ei) =

n∑i=1

( f (end(ei))− f (start(ei))) = 0

as γ is assumed to be a closed path.

”⇐”: Fix a vertex p ∈ V. For q ∈ M define

f (q) =∫

γω

where start(γ) = p and end(γ) = q. This is well defined as weassume M to be connected and by the hypothesis. Now it remains tocheck that d f = ω.


If M is a compact oriented n-dimensional manifold, then we have a naturalmap

Ωk(M)×Ωn−k(M)→ R

given by

(ω, η) 7→∫

Mω ∧ η.

This is a non-degenerate pairing of vector spaces in the sense that∫M

ω ∧ η = 0 for all η ∈ Ωn−k(M) ⇒ ω = 0

and ∫M

ω ∧ η = 0 for all ω ∈ Ωk(M) ⇒ η = 0.

Consider the case that both, ω and η are closed, i.e. dω = dη = 0. For anyother (n− k− 1)-form α ∈ Ωn−k−1(M) we obtain∫

Mω ∧ (η + dα) =

∫M

ω ∧ η − (−1)k∫

Md (ω ∧ η) =

∫M

ω ∧ η

where the last equality is due to Stoke’s theorem. Denoting the exteriorderivative

Ωk(M)dk−→ Ωk+1(M)

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Definition 7.24 (k-th cohomology of M). We define a real vector spacecalled the k-th cohomology of M by

Hk(M) := ker dkImdk−1.

Further〈[ω], [η]〉 :=

∫M

ω ∧ η

is well-defined. Later we will see that

βk(M) := dim Hk < ∞

which will be the k-th betty number of M. We will see that βn−k = βk whichis then the Poincare duality.The problem to define an analogue in the discrete setting is that there isno ”good” wedge product for discrete forms. But still there is a sensiblenotion of a Poincare-duality.

For surfaces the solution will be to bring the dual surface into the picture.There is indeed a good discrete definition of∫

Mω ∧ η

for ω ∈ Ωk(M) and η ∈ Ωn−k(M∗). The strategy will be to stick to theoriginal primal cell-decomposition but to give a suitable interpretation fordiscrete k-forms on M∗.

View f ∈ Ω0(M∗) as a function taking a fixed value fϕ ∈ R on the interiorof each face ϕ ∈ F. Then actually f is the limit of smooth functions whichwe can be imagined as seen in the sketch.

99


Recall that for ω ∈ Ω2(M) we considered an equivalence class of 2-formswhere only

ωϕ =∫

ϕω

mattered (in the sense that we only consider equivalence classes of smooth2-forms which are equal on oriented edges). Now we pair a primal 2-formand a dual 0-form to

〈 f , [ω]〉 :=∫

Mf ω.

From the purely discrete viewpoint this is f ∈ Ω0(M∗) gives fϕ ∈ R for each ϕ ∈ F.ω ∈ Ω2(M) gives ωϕ ∈ R for each ϕ ∈ F.

How do we do this the other way round?

Consider η ∈ Ω2(M∗) as the limit of smooth 2-forms on M.A discrete f ∈ Ω0(M) is an equivalence class of smooth functions whereonly the values f (v) on the vertices v ∈ V matter. Then if

limt→0

∫Uv

ηt = ηv ∈ R

and supp ηt shrinks to v we have

limt→0

∫M

f ηt =∑v∈V

f (v)ηv.

Discrete 1-forms ω ∈ Ω1(M∗) are limits of smooth 1-forms such that

limt→0

∫γ

ωt = ωe.

for all curves γ from the right of e to its left.

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Consider a smooth η ∈ Ω1(M) and ω ∈ Ω1(M∗). With ωt = ft(x, y)dy andη = adx + bdy we obtain

limt→0

∫ ε

−εft(x, y)dy = ωe

for all x ∈ [0, 1] hence

limt→0

∫M

η∧ωt = limt→0

∫ 1

0

∫ ε

−εa(x, y) ft(x, y)dydx =

∫ 1

0a(x, 0)ωe = ωe

∫ 1

0η = [η]e

From the discrete point of view that is: For ω ∈ Ω1(M∗) given by ωe ∈ R

for each e ∈ E with ω$(e) = −ωe and a smooth η ∈ Ω1(M) we set

〈[η], ω〉 := 12

∑e∈E

ηeωe = ”∫

Mη ∧ω”.

So we can think of ω as kind of ”delta-distribution form”.

7.6 ”Baby Riemann Roch Theorem”

In this section we will use M to denote a smooth, oriented, compact andconnected manifold. We fix a cell decomposition M of M by

ÄE, $, s

ä. Note

that we change the notation from σ to s (as in ”shift”). M∗ will denote thediscrete surface dual to M. On M we have

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Ω2(M) Ω1(M) Ω0(M)

Ω0(M) Ω1(M) Ω2(M)

d1 d0

d0 d1

where↔ means that these are paired in the sense that for ω ∈ Ωk(M) andη ∈ Ωn−k(M) we have a non-degenerate

〈ω|η〉 :=∫

Mω ∧ η.

For f ∈ Ω0(M) and ω ∈ Ω1(M) we can pair ω and d0 f to

〈ω|d0 f 〉 =∫

Mω ∧ d0 f =

∫M−d0 f ∧ω

=∫

M−d1( f ω) + f d1ω =

∫M

f d1ω = 〈d1ω| f 〉

where we used the product rule and Stoke’s theorem. This shows that inthe smooth picture

d1 = d∗0 .

For some f ∈ Ω0(M) and ω ∈ Ω1(M) we see that

〈 f |d1ω〉 =∫

Mf d1ω =

∫M

d1( f ω)− d0 f ∧ω = − 〈d0 f |ω〉

henced∗1 = −d0.

Remark 7.25. One can change the pairings in a way such that the − sign inthe last equation vanishes.

In the discrete setting the diagram becomes

Ω2(M∗) Ω1(M∗) Ω0(M∗)

Ω0(M) Ω1(M) Ω2(M)

∂1 ∂0

d0 d1

The definition was that for f ∈ Ω0(M)

d f (e) = fend(e) − fstart(e)

henced f (e) = 0 ⇔ f = const.

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In particular,dim ker d0 = 1.

From the theorem that M∗ is also connected if M is connected we immedi-ately get that also

dim ker ∂0 = 1.

The kernel thus consists of the constant functions. An interesting conse-quence is the following:

Theorem 7.26. Let σ ∈ Ω2(M) be a discrete 2-form. Then there is a discrete1-form ω ∈ Ω1(M) such that dω = σ (i.e. σ is exact) if and only if∫

Mσ = 0.

Proof. The first implication follows directly from Stoke’s theorem, whichholds also in the discrete setting. The contrary implication follows fromlinear algebra facts, because im d1 = (ker d∗1)

⊥. This gives

im d1 =ß

σ ∈ Ω2(M) |∫

Mσ = 〈1|σ〉 = 0

™so that we are done.

Remark 7.27. The theorem also holds for smooth manifolds, but the proofrequires knowledge about elliptic operators which is beyond the scope ofthis lecture.

Theorem 7.28. It holds that d∗0 = ∂1 and d∗1 = −∂0.

Proof. The first identity will be a homework exercise. For the second iden-tity we compute on the one hand that

〈 f ∗|dω〉 =∑ϕ∈F

fϕdωϕ =∑ϕ∈F

fϕ

∑left(e)=ϕ

ωe.

On the other hand we have

− 〈∂ f ∗|ω〉 = −12

∑e∈E

Äfright(e) − fleft(e)

äωe

= 12

∑e∈E

fleft(e)ωe − 12

∑e∈E

fright(e)ωe

= 12

∑ϕ∈F

∑left(e)=ϕ

fϕωe − 12

∑ϕ∈F

∑right($(e))=ϕ

fright($(e))ω$(e)

=∑ϕ∈F

∑left(e)=ϕ

fϕωe

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So we have already found out that by the dimension formula

dim im d1 = |F| − 1 = |E| − dim ker d1.

This givesdim ker d1 = |E| − |F|+ 1

which is the number of closed 1-forms on M. Analogously one can derivethat

dim im d0 = |V| − dim ker d0 = |V| − 1

where we sue that dim ker d0 = 1. Combining these we get

β1(M) := dim H1(M) = dimÅ

ker d1im d0

ã= 2− χ(M).

Definition 7.29 (First Betti-number). We define the first Betti-number of adiscrete surface M to be

β1(M) := dim H1(M).

Remark 7.30. Note that by the Poincare-Hopf index theorem, χ(M) =χ(M) and therefore there is no typo in the equations above and the Betti-number does not depend on the cell-decomposition of M.

Theorem 7.31 (”Baby” Riemann-Roch). The first Betti-number of M is givenby

β1(M) = 2− χ(M).

Proof. This is the result of the observations above.

To obtain a smooth version of the Riemann-Roch theorem we need to getrid of the ˆ in β1(M). Consider the map

Ωk(M)π−→ Ωk(M), ω 7→ ω.

Then

Ω0(M) Ω1(M) Ω2(M)

Ω0(M) Ω1(M) Ω2(M)

d0 d1

d0 d1

π0 π1 π2

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which is by Stoke’s theorem a commutative diagram. By some diagram-chasing we get that π1(im d0) ⊂ im d0 and π1(ker d1) ⊂ ker d1 and thus awell defined map

$ : ker d1im d0︸︷︷︸=H1(M)

−→ ker d1im d0︸︷︷︸=H1(M)

, [ω] 7→ $([ω]) := [ω].

Since π1 is surjective the map $ is surjective.

Lemma 7.32. The map $ is also injective.

Proof. Let ω ∈ Ω1(M) and dω = 0. As we assumed the cell-decompositionto be diffeomorphic to convex sets there are (by the Poincare Lemma)

gv : Uv → R such that dgv = ω|Uv .

These are unique up to an additive constant. Further $([ω]) = 0 meansthat ω is exact, i.e. ω = d f for some f ∈ Ω0(M). We can interpret f asa function on M which is constant on the dual faces Uv. As an exercisecheck that ω = d(g− f ) which then yields that [ω] = 0, hence yields theclaim.

7.7 A Natural Complex Structure on Dual Spaces

We want to answer the following question:

”Let V be a vector space with complex structure J. What is the natural complexstructure on V∗?”

More generally, if f : V → W is an isomorphism of vector spaces, how dolinear forms ω ∈ V∗ transform under f ?

We are looking for f : V∗ →W∗ such that¨f (α), f (v)

∂= 〈α, v〉 .

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7.8 Holomorphic Structures on Vector Bundles

By rewriting the equality using the adjoint map we get¨f ∗ f (α), v

∂=¨

f (α), f (v)∂= 〈α, v〉

which implies that f = ( f ∗)−1.

Applying the observations above to f = J yields that the natural complexstructure on V∗ is given by

(J∗)−1 =Ä

J−1ä∗ = −J∗.

Definition 7.33 (Hodge-star). For a Riemann-surface M, we have a complexstructure J on TM. This gives rise to a complex structure ∗ = −J∗ on T∗M,the so called Hodge-star.

Remark 7.34.

i) We have that ∗ ∈ ΓEnd(T∗M) and if ω ∈ Ω1(M), then for X ∈ Γ(TM)it holds that

−ω(JX) = ∗ω(X),

because

∗ω(X) = 〈∗ω|X〉 = 〈−J∗ω|X〉 = − 〈ω|JX〉 = −ω(JX)

ii) There are many papers (including many that originate from Berlin,or the C-seminar by Franz Pedit) which use what we refer to as the”Berlin convetntion” which is

∗ω(X) = ω(JX),

i.e. ∗ = J∗. But this does only work for surfaces and causes trouble inhigher dimensions.


We have a natural splitting of HomR(TM, C) = T∗M⊗C into

HomR(TM, C) = K⊕ K,

where

K := ω ∈ HomR(TM, C) | ω(JX) = Jω(X) for all X ∈ Γ(TM)= ω ∈ HomR(TM, C) | ∗ω = −Jω

and

K := ω ∈ HomR(TM, C) | ω(JX) = −Jω(X) for all X ∈ Γ(TM)= ω ∈ HomR(TM, C) | ∗ω = Jω .

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Definition 7.35 (canonical bundle). The bundle K that arises from the abovesplitting is called the canonical bundle of M.

For a line bundle L we denote the bundle whose fiber at p ∈ M is L∗Cp

by L−1. Here ∗C has to be understood as the complex dual space, i.e.HomC(L, C). As the line bundle takes values in C, any nonzero z ∈ C

has a natural inverse z−1. Also we note that the L−1 is indeed the samewe know from the classification of line bundles. In this notation one oftencomes across (in particular in pure complex analysis literature) the termTM = K−1.

Definition 7.36 (canonical bundle splittings). For a complex vector bundleE over a Riemann surface M we define

KE := ω ∈ HomR(TM, E) | ω(JX) = Jω(X) for all X ∈ Γ(TM)

and

KE := ω ∈ HomR(TM, E) | ω(JX) = −Jω(X) for all X ∈ Γ(TM) .

Using the canonical bundle splitting we can decompose any ω ∈ Ω1(M, E)into

ω = ω′︸︷︷︸∈Γ(KE)

+ ω′′︸︷︷︸∈Γ(KE)

so that more generally

Ω1(M, E) = Γ(KE)⊕ Γ(KE).

If E is further equipped with a complex connection∇ then we can introducea splitting of the 1-form ∇ψ into

∇ψ = ∂ψ + ∂ψ.

Thus in the notation above ∂ψ corresponds to ω′ and ∂ψ to ω′′ respectively.To summarize: Let M be a Riemann surface and E a complex line bundleover M with connection ∇. Then we can define ∂ : Γ(E)→ Γ(KE) by

∂ψ := (∇ψ)′′ := 12(∇ψ− J ∗ ∇ψ) .

In particular, for the trivial bundle E = M × C with trivial connection∇ = d we have Γ(M×C) = C∞(M, C) (”more or less the same thing as”)and

d f = 12 (d f + J ∗ d f ) + 1

2 (d f − J ∗ d f ) .

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We check that these are indeed contained in the rightful spaces by comput-ing

∗∂ f = 12 (∗d f − Jd f ) = −J∂ f

∗∂ f = . . . = J∂ f .

Definition 7.37 (holomorphic sections). Let ∇ be a complex connection ona complex vector bundle E. Then ψ ∈ Γ(E) is called holomorphic if ∇ψ ∈Ω1(M, E) is complex linear at all points, i.e. ∂ψ = 0.

For f ∈ C∞(M, C) holomorphicity means

d f (JX) = id f (X)

which, spelled out, is nothing more than the Cauchy-Riemann equations.

Definition 7.38 (holomorphic structure). A complex linear map

∂ : Γ(E)→ Γ(KE)

is called a holomorphic structure (or ”∂-operator”) if

∂( f ψ) = (∂ f )ψ + f ∂ψ

holds for all f ∈ C∞(M) and ψ ∈ Γ(E).

Remark 7.39. Two things have to be noted. Sometimes one finds the nomen-clature ∂-operator instead of holomorphic structure. Also we solely need toask for real valued C∞(M) functions f to satisfy the product rule. We getthe complex valued functions f ∈ C∞(M, C) for free as we can compute forf = u + iv that

∂( f ψ) = ∂(uψ) + J∂(vψ) = (∂u)ψ + u∂ψ + J(∂v)ψ + Jv(∂ψ)

= ∂(u + iv)ψ + (u + iv)∂ψ = (∂ f )ψ + f ∂ψ.

It remains to check if ∂ operator that arise from complex connections areindeed holomorphic structures.

Proposition 7.40. If ∂ comes from a complex connection ∇ on E, then ∂ is aholomorphic structure.

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Proof.

∂( f ψ) = 12 (∇( f ψ)− J ∗ ∇( f ψ))

= 12 (d f ψ + f∇ψ− J ∗ (d f ψ + f∇ψ))

= 12 (d f − J ∗ d f )ψ + f 1

2 (∇ψ− J ∗ ∇ψ)

= (∂ f )ψ + f (∂ψ)

Theorem 7.41. If ∂ is a holomorphic structure on E and ω ∈ ΓÄKEndCE

äthen ∂ + ω is another holomorphic structure and all holomorphic structures onE arise in this way.

Proof. The simple computation

(∂ + ω)( f ψ) = (∂ f )ψ + f (∂ψ) + f ωψ = (∂ f )ψ + f (∂ + ω)ψ

yields the first part of the theorem. If ∂1 and ∂2 are two holomorphic struc-tures on E then one can compute that

(∂2 − ∂1)( f ψ) = f (∂2 − ∂1)ψ

which shows that (∂2 − ∂1) is a tensor.

In particular for a line bundle L with holomorphic structure ∂ it is EndCL ∼=M × C. So ∂ω for some ω ∈ ΓK parametrizes all holomorphic structureson L.

7.9 Elliptic Differential Operators

Definition 7.42 (first order differential operator). Let E, F be vector bun-dles over a manifold M. Then a linear map D : Γ(E) → Γ(F) is called a firstorder differential operator, if there is A ∈ ΓHom(TM∗, Hom(E, F)) suchthat, for all ψ ∈ Γ(E), f ∈ C∞(M),

D( f ψ) = Ad f ψ + f Dψ .

A ∂-operator is a first order differential operator: Here A is the projectionon the complex anti-linear part, i.e.

A : ω 7→ aω = ω′′.

109


Let us look at this locally: Choose a local frames ϕ1, . . . , ϕk on E andϕ1, . . . , ϕk on E, say on the open neighborhood U. Then a section ψ ∈ Γ(E)can be written locally as

ψ =∑

yi ϕi, yi ∈ C∞(U) .

Let further x` denote some coordinate functions of M on U. Then

Dψ =∑

(Adyi ϕi + yiDϕi) =∑

i(∑`

∂yix`

Adx` ϕi + yiDϕi) =∑i,j,`

( ∂yix`

bij`+ yiaij)ϕj

A is uniquely determined by D and we make the definition:

Definition 7.43 (symbol of a differential operator). Consider a first orderdifferential operator D, then the uniquely determined map A as from above iscalled the symbol of D.

Let M be a compact oriented n-dimensional manifold and E a vector bundleover M. For ψ ∈ Γ(E) and ω ∈ Ωn(M, E∗) we define

〈〈ω | ψ〉〉 :=∫

M〈ω|ψ〉 .

This yields a non-degenerate pairing.

Definition 7.44 (adjoint map). If we have non-degenerate pairings betweenV and V and between W and W and we have a linear map B : V → W, thena linear map B : W → V is called an adjoint of B if for all v ∈ V and w ∈ Wwe have

〈w|Bv〉 =¨Bw

∣∣∣v∂ .

Note that B (if it exists) is unique. So we write B = B∗.

In gereral one can prove the following theorem—which is basically integra-tion by parts.

Theorem 7.45. If E, F are vector bundles over a compact oriented manifold Mand

D : Γ(E)→ Γ(F)

is a first order differential operator, then D has an adjoint

D∗ : Ωn(M, F∗)→ Ωn(M, E∗)

and D∗ is again a first order differential operator.

Fortunately this theorem is almost never needed in practice because D∗ canbe written down explicitly in the concrete case.

110


Definition 7.46 (elliptic operator). A first order differential operator

D : Γ(E)→ Γ(F)

is called elliptic if for all 0 6= ω ∈ T∗p (M), p ∈ M the linear map

Aω : Ep → Fp

is a vectorspace isomorphism.

Theorem 7.47 (The elliptic theorem). Let E, F are vector bundles over a com-pact oriented manifold M and D : Γ(E)→ Γ(F) is a elliptic first order differen-tial operator. Then:

i) dim ker D < ∞.

ii) D∗ : Ωn(M; F∗) → Ωn(M; E∗) is also elliptic and im D = (ker D∗)⊥.In particular, (im D)⊥ = ker D∗.

Definition 7.48 (dual pairing). Let M be a compact oriented n-dimensionalmanifold and E, E be vector bundles over M. Then a dual pairing between Eand E is a tensorial bilinear map

〈.|.〉 : Γ(E)× Γ(E)→ ΩnM

such that 〈ω|ψ〉p = 0 for all ψ implies that ωp = 0 and vice versa.

In particular

〈〈ω|ψ〉〉 :=∫

M〈ω|ψ〉

defines a non-degenerate pairing between Γ(E) and Γ(E), so Γ(E) may beviewed as (Γ(E))∗.

Given a dual pairing between E and E yields a vector bundle isomorphism

E→ Ωn(M; E∗), ω 7→ ω

where〈ω(X1, . . . , Xn)|ψ〉 = 〈ω|ψ〉(X1, . . . , Xn) .

Example 7.49. E = ΛkTM∗ and E = Λn−kTM∗ have a dual pairing

〈ω|η〉 = ω ∧ η .

111


On a Riemannian manifold the Hodge-star operator

∗ : ΛkTM∗ → Λn−kTM∗

is given byω ∧ ∗η = 〈ω, η〉det .

It satisfies ∗∗ = (−1)k(n−k). The Hodge-star yields a dual pairing of ΛkTM∗

with itself. This pairing extends to the exterior algebra E = ΛTM∗ such thatforms of different degree pair to zero.

Let ΩM = ΓE. The exterior derivative

d : ΩM→ ΩM

is a first order differential operator with symbol

Ad f (ω) = d f ∧ω,

henced( f ω) = d f ∧ω + f dω .

It has an adjoint which is d∗ = δ = ± ∗ d∗ where the sign depends ondimension. E.g. in case n = 2 one may check that d∗ = δ = − ∗ d∗.Moreover,

δ( f ω) = εk ∗ (d f ∧ ∗ω) + f δω .

Theorem 7.50. D : ΩM→ ΩM, D = d + δ is a self-adjoint elliptic first orderoperator, called the Hodge-Dirac operator.

Proof for n = 2. D( f ψ) = f Dψ + d f ∧ ψ− ∗(d f ∧ ∗ψ). Thus

Ad f ω = d f ∧ω− ∗(d f ∧ ∗ω) .

For 0 6= η ∈ TpM∗,

Aη

Öfω

g det

è=

Ö− ∗ (η ∧ ∗ω)

f η − g ∗ ηη ∧ω

è⇒

η ∧ω = 0η ∧ ∗ω = 0

f η − g ∗ η = 0⇒

f = 0ω = 0g = 0

Thus Aη bijective.

Theorem 7.51. Let M be a connected n-dimensional compact oriented manifold.Then

σ ∈ ΩnM exact⇐⇒∫

Mσ = 0 .

112


Proof. ⇒ follows from Stokes’ theorem. ”⇐”: Let∫

Mσ = 0. If ( f , ω, g det) ∈

ker D, then

0 =∫

M(d f − ∗dg) ∧ (∗d f + dg) = ‖d f ‖2 + ‖dg‖2

and hence f and g are constant. In particular, (0, 0, σ) ∈ (ker D)⊥ = im D.Thus σ is exact.

From now on let M denote a compact Riemannian surface, E a complexvector bundle over M and L a complex line bundle over M. Let ∂ be aholomorphic structure on E:

∂ : ΓE→ ΓKE, ∂( f ψ) = (∂ f )ψ + f ∂ψ .

If ω ∈ T∗M then Aωψ = ω′′ψ. In particular, if ω 6= 0, then ω′′ 6= 0 andhence ψ 7→ Aωψ is bijective.

Example 7.52. 1. If ∇ is a connection on E, then ∇ = ∇′ + ∇′′ and∂∇ = ∇′′ is a ∂-operator.

2. Let E = M×C, ∇ = d, then ∂∇ = ∂ is the canonical ∂ on C∞(M; C).

3. Consider E = TM and let I be the identity (tautological 1-form). Thend∇ I = T∇ – the torsion of the connection. Suppose a complex con-nection ∇ on TM is torsion-free. Then

∂∇X Y = 12(∇XY + J∇JXY) = 1

2(∇XY−∇YX + J[JX, Y])

= 12([X, Y] + J[JX, Y]) .

The last example is worth to be made into a theorem.

Theorem 7.53. All torsion-free complex connections ∇ on TM have the same∂∇ =: ∂ given by

∂XY = 12([X, Y] + J[JX, Y]) .

What about KL = Hom+(TM, L) if L comes with a complex connection ∇?Again, let ∇M be a torsion-free complex connection on TM. Then

(∇Xω)(Y) := ∇X(ω(Y)−ω(∇MX Y)

defines a complex connection. Then

(∂∇X ω)(Y) = 12

Ä∇X(ω(Y)−ω(∇M

X Y) + J∇X(ω(Y)− Jω(∇MX Y)

ä= ∂∇X (ω(Y))−ω(∂XY) .

113


The exterior derivative of ω ∈ Ω1(M; L) is given by

d∇ω(X, Y) = ∇Xω(Y)−∇Yω(X)−ω([X, Y]).

For Y ∈ ΓTM we have d∇ω(., Y) ∈ Ω1(M; L) and so (d∇ω(, .Y))′′ ∈ ΓKL.Thus

(d∇ω(., Y))′(X) = 12

Ä∇Xω(Y)−∇Yω(X)−ω([X, Y])

+ J(∇JXω(Y)−∇Yω(JX)−ω([JX, Y]))ä

= ∂∇X (ω(Y))−ω(∂XY) .

In particular, if L = M×C and ∇ = d:

(∂Xω)(Y) = ∂X(ω(Y))−ω(∂XY) = (d∇ω(., Y))′′(X) .

Behind this fact, there is some identification: To each σ ∈ Ω2(M; L) we canassign σ

∫Γ(K(KL) given by

σX(Y) = 12(σ(X, Y) + Jσ(JX, Y)) .

One may check that σ is of type K in the second slot:

σX(JY) = 12(σ(X, JY) + Jσ(JX, JY)) = 1

2(−σ(JX, Y) + Jσ(X, Y)) = JσX(Y) .

Theorem 7.54. The map

Φ : Λ2(M, L)→ K(KL), σ 7→ σ

is an isomorphism of complex line bundles such that Φ d∇ = ∂.

The fact that Φ d∇ = ∂ is equivalent to saying that the following diagramcommutes:

KL K(KL)

Λ2(M, L)

d∇

∂

σ 7→ σ

Theorem 7.55. ω ∈ ΓK is closed⇔ ω holomorphic.

Remark 7.56. Note that if L is a holomorphic line bundle (i.e. L comes with∂ ), then we automatically get ∂ on KL by the identification above.

114


Theorem 7.57. If L, L are holomorphic line bundles. Then there is a uniqueholomorphic structure on L⊗ L such that

∂(ψ⊗ ψ) = ∂ψ⊗ ψ + ψ⊗ ∂ψ

Proof. If locally f , g ∈ C∞(U), f · g = 1, then...

Let M be a compact Riemann surface of genus

g = 1− 12 χ(M)

and L→ M be a holomorphic line bundle.

Definition 7.58.

(i) A section ψ ∈ Γ(L) is called holomorphic if ∂ψ = 0.

(ii) H0(L) := ker ∂ denotes the set of all holomorphic sections of L.

(iii) h0(L) := dim H0(L).

Theorem 7.59 (Riemann–Roch theorem). Let M be a compact Riemann sur-face of genus g = 1− 1

2 χ(M) and L→ M be a holomorphic line bundle. Then

h0(L)− h0(KL−1) = deg L + 1− g .

All zeros of a holomorphic section have positive index. Hence we immedi-ately obtain the following theorem.

Theorem 7.60. If deg L < 0, then h0(L) = 0.

Corollary 7.61. deg L > 2g− 2, then h0(L) = deg L + 1− g.

Let us first prove the Riemann–Roch theorem for the case that L is the trivialbundle:

ΓK

C∞(M, C)

ΓK

Ω2(M, C)

∂

d

115


We have the diagramm above where the dashed arrows implicate identifi-cation.For σ ∈ Ω2(M; C), f ∈ C∞(M; C) we have a pairing

〈σ| f 〉 =∫

Mf σ .

Similarly, for ω ∈ ΓK, η ∈ ΓK we have a pairing

〈ω|η〉 =∫

Mω ∧ η .

Since

ω ∧ η(X, JX) = ω(X)η(JX)−ω(JX)η(X) = −2Jω(X)η(JX).

we find that this pairing is non-degenerate.

Theorem 7.62. If M is connected, then h0(M × C) = 1, i.e. only constantfunctions are holomorphic.

The proof uses the following hermitian inner product on ΓK:

〈ω, η〉 = 12i

∫M

ω ∧ η .

This means that

〈λω, η〉 = λ 〈ω, η〉 and 〈ω, λη〉 = λ 〈ω, η〉

and we see that

〈ω, η〉 = − 12i

∫M

ω ∧ η = 12i

∫M

η ∧ω = 〈η, ω〉 .

Proof. Clearly constant functions are holomorphic. We show that any holo-morphic function is constant. So suppose that ∂ f = 0. Then

0 = d2 f = d(∂ f + ∂ f ) = d∂ f .

Using Stokes’ theorem, we get

〈∂ f , ∂ f 〉 = 12i

∫M

∂ f ∧ ∂ f = 12i

∫M

∂ f ∧ d f = 12i

∫M

f d∂ f = 0

Hence ∂ f = 0, so d f = 0 and f must be constant.

Theorem 7.63. σ ∈ Ω2(M; C) exact⇐⇒∫

Mσ = 0.

116


Proof.

σ ∈ im (d : ΓK → Ω2(M; C))⇐⇒ σ ⊥ ker (∂ : C∞(M; C)→ ΓK) = C · 1

⇐⇒ 0 = 〈σ|1〉 =∫

Mσ .

Theorem 7.64 (Elliptic theorem - a Riemann-Roch version for L = M×C

on ∂-problem). Let η ∈ ΓK. Then the ∂-problem

∂ f = η

is solvable if and only if η ⊥ H0(K).

Proof. To see this note that, for f ∈ C∞(M; C) and ω ∈ ΓK,

〈∂ f |ω〉 =∫

M∂ f ∧ω =

∫M

d f ∧ω = −∫

Mf dω .

Hence d = −∂∗. The minus sign is not important for image and kernel.

Again, let us consider the following diagram:

ΓK

C∞(M, C)

ΓK

Ω2(M, C)

∂

d

The pairings are given by

〈〈ω|η〉〉 =∫

Mω ∧ η, ω ∈ ΓK, η ∈ ΓK ,

〈〈 f |σ〉〉 =∫

Mf σ, f ∈ C∞(M; C), σ ∈ Ω2(M; C) .

We have

〈〈 f |dω〉〉 =∫

Mf dω = −

∫M

d f ∧ω = −∫

M∂ f ∧ω =

∫M

ω ∧ ∂ f = 〈〈ω|∂ f 〉〉 .

Thusd = ∂∗ .

117


Let us keep in mind the following calculation: On M = C we have dz ∈ ΓK.dz = dx + idy. Then dz ∧ dz = 2idx ∧ dy and so dx ∧ dy = 1

2i dz ∧ dz. Then apositive hermitian product 〈., .〉 on ΓK is given by

〈〈ω, η〉〉 = 12i 〈〈ω|η〉〉 .

By the elliptic theorem we have that

Im ∂ = (H0K)⊥, Im d = (C1)⊥ .

Recall that

2g = dimR

ker(d : Ω1(M; R)→ Ω2(M; R))

im(d : C∞(M; R)→ Ω1(M; R)).

To each α ∈ Ω1(M; R) there corresponds ω ∈ Γ(K) given by

ω = 12(α + iα J) = α′′ ∈ ΓK .

The harmonic forms are those which are closed and co-closed:

harm (M) := α ∈ Ω1(M; R) | dα = d ∗ α = 0 .

Obviously, α ∈ harm (M)⇔ α′′ ∈ H0M and ω ∈ H0M⇔ Reω ∈ harm (M).

Theorem 7.65 (Hodge–theorem). Let η ∈ Ω1(M; R) be closed, dη = 0. Thenthere is a unique α ∈ harm (M) and f ∈ C∞(M; R) (unique up to a constant)such that

η = α + d f .

Proof. Let ω1, . . . , ωk be a basis of H0(K) such that ω1, . . . , ωk ∈ ΓK areorthonormal. Then η := η′′ −

∑j〈〈ωj, η〉〉ωj will satisfy 〈ωj, η〉 = 0 for all j,

i.e. η ⊥ ker d, and by the elliptic theorem we get η ∈ im ∂. So η = ∂ f forf ∈ C∞(M; C). Let α′′ :=

∑j〈〈ωj, η′′〉〉ωj. To α′′ corresponds a real-valued

harmonic form α ∈ harm (M). Now,

η′′ = ∂ f + α′′ = 12(d f − i ∗ d f ) + 1

2(α− i ∗ α) .

If we take the real part then we get, with f = u + iv,

η = du + ∗dv + α .

Since dη = 0, we obtain d ∗ dv = 0 and hence v must be a constant:

0 = −∫

Mvd ∗ dv =

∫M

dv ∧ ∗dv = ‖dv‖2

118


In other words: each homology class [η] ∈ H1(M) contains exactly oneharmonic representative α. Thus

2g = dimR H1(M) = dimR harm (M) ; g = dimC H0(M)

where 2g− 2 = deg K.

Theorem 7.66. Every complex holomorphic line bundle of degree zero over Mhas a flat connection ∇ with ∇′′ = ∂.

Proof. Choose one connection ∇ with ∇′′ = ∂ and look for ω ∈ ΓK suchthat ∇ = ∇+ iω has curvature R = R+ idω = i(dω−Ω) = 0, i.e. dω = Ω,which is solvable by elliptic theorem because 0 = 2πdeg (L) =

∫M

Ω.

Corollary 7.67. Every holomorphic line bundle over a (not necessarily compact)Riemann surface M locally has a holomorphic section without zeros.

Proof. Glue the bundle locally in a degree zero bundle and use the previoustheorem.

Theorem 7.68. Any almost complex surface is complex.

Proof. Locally there is a holomorphic basis section ω ∈ ΓK, dω = 0. Sinceclosed forms are locally exact, there is a locally defined complex-valuedfunction z such that ω = dz. z is a holomorphic chart.

Corollary 7.69. Let L be a holomorphic line bundle over a Riemann surfaceM. Then, expressed in a local holomorphic chart z : U → C and using a localholomorphic basis section ϕ, any holomorphic section ψ is of the form ψ = f (z)ϕfor some holomorphic function f : z(U)→ C.

Thus things locally behave as expected. We can define meromorhic sections,removable singularities, etc. also for holomorphich sections of holomorphicline bundles.

Theorem 7.70 (Riemann–Roch). Let L be a holomorphic line bundle over acompact Riemann surface M of genus g. Then

h0(L)− h0(KL−1) = deg L− g + 1 .

This is a special case of the Atiyah–Singer index theorem on the index ofan elliptic operator. Recall again the situation.

119


ΓKL−1

ΓL

ΓKL

Ω2(M; L)

∂

d∇

We break this into several lemmata.If p ∈ M there is a holomorphic line bundle (p) of degree 1 which has acanonical section ψ ∈ H0(L) with only one simple zero at p—the skyscraperbundle. If L and L are two holomorphic line bundles with sections ψ ∈ H0Land ψ ∈ H0 L with only one simple zero p. Then ρ(zψ) = zψ determines anisomorphism of holomorphic bundles—on all of M.

Lemma 7.71. Riemann–Roch holds for L⇔ Riemann–Roch holds for L(p).

Which, by h0(L(p))− h0(KL−1(−p)) = deg L+ 1− g+ 1, itself follows from

Lemma 7.72. h0(L(p))− h0(L) = h0(KL−1(−p))− h0(KL−1) + 1

Proof. Let ϕ ∈ Γ(p) be the canonical holomorphic section of (p). If ψ isa meromorhic section of L with only a simple pole at p and otherwiseholomorphic. Then ψϕ ∈ H0(L(p)). Conversely, if ψ ∈ H0(L(p)), thenψ/ϕ has at most a simple pole at p. Thus

h0(L(p)) = dimψ ∈ H0(L|M\0) | at most a simple pole at p .

If h0(L(p)) > h0(L), then there is ψ ∈ H0(L(p)) with ψp 6= 0—otherwisedivision by ϕ would yield an injective map H0(L(p))→ H0(L).Now suppose we would have h0(KL−1(−p)) > h0(KL−1). Then we wouldhave an L−1-valued meromorphic 1-form ω with a simple pole in p.

h0(L(−p)) = dimψ ∈ H0(L) | ψp = 0 .

If ψp ∈ H0(L), ψp = 0, then ψϕ−1 ∈ H0(L(−p)), where ϕ−1 denotes thecanonical section of (−p).Then 〈ω|ψ〉 meromorphic C-valued 1-form on M with a single simple poleat p, which cannot exists by the residue theorem (residues must sum upto zero — follows immediately from Stokes’ theorem). Hence we have

120


h0(KL−1(−p)) = h0(KL−1). So, if h0(L) goes up by tensoring in (−p), thenh0(KL−1) does not. Now

L := KL−1(−p), L(p) = KL−1, KL−1 = KK−1L(p) = L(p), KL−1(−p) = L .

The same arguments as before applied to L, yields then that if h0(KL−1)does not go up, then h0(L) goes up.The dimension can increase at most by one (why?). So we have

h0(L(p))− h0(L) ∈ 0, 1, h0(KL−1)− h0(KL−1(−p)) ∈ 0, 1 .

Furthermore,

h0(L(p))− h0(L) = 1⇔ not (h0(KL−1(−p))− h0(KL−1) = 1).

It follows that h0(L(p))− h0(L) = 1− h0(KL−1) + h0(KL−1(−p)).

If we keep tensoring in (p), the bundles on the right hand side becomeof negative degree. Hence after a while, h0(L(kp)) > 0. Hence L has ameromorphic section with a k-order pole at p.

Theorem 7.73. Every holomorphic line bundle over a compact Riemann surfaceM has a meromorphic section.

Tensoring in the point bundles corresponding to the zeros and poles ofthis meromorphic section we arrive at the trivial bundle. So we can applyRiemann–Roch for this case.

121

8. Appendix

The aim of this appendix is to give a short overview over additional top-ics of interest concerning complex analysis and Riemann surfaces. In thefirst place the plan will be to summarize the main ideas that are discussedwithin the turtorials. Some of the presented summaries are due to TheoBraune who volunteered to take on the burden to type them.

On the Stereographic Projection

Vividly, the stereographic projection from the north pole σN : S2 \ N → C,or the south pole σS : S2 \ S → C respectively, provides us with a bijec-tion of the punctured Riemann sphere to C. It now may be interesting tohave an explicit formula for the map for various reasons, one of which willbe pointed out soon.

Consider for example σN, then we can parameterize the straight line alongwhich we project, say γ, by

γ(t) = tN + (1− t)P = t

Ö001

è+ (1− t)

Öxyz

è.

Then γ(t) = 0 yields t = −z1−z thus 1 − t = 1

1−z . Hence we see that incartesian coordinates the map is given by

σN : S2 \ N → C,

Öxyz

è7→ x

1− z+ i

y1− z

.

In a similar manner we can conclude that

σS(x, y, z) =x

1 + z+ i

y1 + z

.

122

Appendix

The respective inverses of σN and σS are given by

σ−1N (z) =

Å2Re(z)1+|z|2

, 2Im(z)1+|z|2

,−1−|z|2

1+|z|2

ã

Considering the two maps σN, σS as charts of S2 then we can ask ourselveshow coordinate changes look like. As an example we take the coordinatechange from σN to σS, then for z ∈ C \ 0 it is

σS σ−1N (z) =

1z

.

It turns out that the same holds for the coordinate change from σS to σN.Moreover, if we choose σS instead of σS, then (σN, σS) form an atlas of S1

and the transition map is σS σ−1N = 1

z and in particular holomoprhic. Asz 7→ 1

z is a holomorphic map, coordinate changes on the Riemann sphereS2 = C are holomorphic what turns it into a Riemann surface.

On Topological Manifolds

In the second chapter, we said that we want the kind of topological spacesthat we deal with to have nice properties and therefore demanded them tobe Hausdorff, second countable and to locally ”look like” euclidean space.To understand why all three of these properties are reasonable to ask for,we now want to come up with examples of topological spaces that fail tofulfill exactly one of them.

Hausdorff: A second countable, locally euclidean space that is not Haus-dorff is for example given by the line with two origins. Define

X = R× a⋃

R× b

123

and define an equivalence relation

(x, a) ∼ (x, b) :⇔ x 6= 0.

The resulting space X∼ then looks like the real line just with twodifferent zeros 0 and 0′.

It is easy to imagine that if we consider both zeros, any open neigh-bourhoods of these will intersect. The details are left as an exercise.

2nd countable: A locally euclidean Hausdorff space that does not fulfill thesecond axiom of countability is given by

X = (R,D)× (R, S)

where D is the discrete topology and S the standard topology on R.A basis of X is given by x × (a, b) | x, a, b ∈ R, a < b. Using theinsight that this indeed is a Basis of X immediately gives an idea howto get a contradiction if we assume the existence of a countable basisof X. Again the details are left as an exercise.

locally euclidean: A standard example for a space that is not locally eu-clidean is given by a curve with one end arbitrary close to its owntrajectory, for example consider

γ : (−2, 1)→ R2, t 7→

(0, |t|) , t ∈ (−2, 0](cos(2πt) + 1, sin(2πt)) , t ∈ (0, 3

4)(1 +

34−t

14

, 1)

, t ∈ [34 , 1)

The trajectory of the curve looks like this:

124

Appendix

Again, the details are left as an exercise, but it is easily imaginablethat it is not locally euclidean at (0, 1).

On Linear Complex Structures

In the lecture we have learned, that a complex structure is determined by thelinear map J : V → V. This is because a map f : R2n → R2m is defined tobe holomorphic if it is continuously differentiable in the real sense and it iscomplex-linear, i.e. d f (Jv) = Jd f (v) holds.Although it looks clear what is going on, the term

d f (JR2n v) = JR2m d f (v)

has the potential to provide trouble, because, as we now have completelywritten it in detail, we see that the abbreviation AJ = JA defrauds that inparticular we have to deal with two different complex structures, one forthe domain and one the range, that have to fit together for holomorphicity.

Consider for example C ∼= R2. If we want to complexify R2, we have tofind a linear map J : R2 → R2 such that J2 = −I. Then we can define

(a + ib)v := (a + bJ)v.

As we already know what the complex structure on C is, namely i whichis geometrically nothing more than a 90-rotation, we can simply make useof the 90-rotation matrix that we know from linear algebra and define

J2 :=ñ0 −11 0

ô,

then this is clearly linear and it satisfies J2 = −I. Also we see the expectedanalogy to the actual complex number, meaning as x + iy↔ [x y]T we havethe identification

i(x + iy) = −y + ix ↔ J2

ñxy

ô=

ñ−yx

ô.

Note that also J2 := −J2 would do the trick. This simply correspnds tochanging the orientation of C. But as a matter of fact, these are the onlypossible choices for the case C ∼= R2 if we demand our complex structureto preserve angles and length. Check for example that for b ∈ R

Jb :=ñ

0 −b1b 0

ô125

also defines a complex structure, but apparently lacks of this property ifb 6= 1.

Let us consider a higher dimensional example: C2 ∼= R4. The questionof how many possible orthogonal 90-rotations there possibly are is muchharder to answer in this setup. It turns out that there in fact are S2-maypossibilities.

Our first approach to get this insight is based on quaternions. We knowthat R4 ∼= H where

H := x0 + ix1 + jx2 + kx3 | x0, x1, x2, x3 ∈ R

with i2 = j2 = k2 = ijk = −1. Choose for example 1 ∈ H, then the orthog-onal complement 1⊥ ∼= R3, because 〈1, i〉 = 〈1, j〉 = 〈1, k〉 = 0. As we aimfor orthogonal, the 2-sphere S2 ⊂ R3 are all quaternions of unit length thatcorrespond to a orthogonal 90-rotation.1

Another suitable Approach that may acquire intuition is illustrated by thepicture below.

Having chosen v ∈ R3, assume |v| = 1 then the orthogonal complement v⊥

of v is the plane where the equator of S2 with respect to v is inscribed. Soany map J that maps v to the equator is an orthogonal 90-rotation.But note that for our actual setup v ∈ R4, thus S2 becomes S3 and thethe equator, as it is an S1 becomes an S2, thus we have S2 many possibleorthogonal 90-rotations, meaning that¶

J ∈ EndR

ÄR4ä | J2 = −1

© ∼= S2.

Here again, allowing scaling factors leads to non-orthogonal complex struc-tures of which we can have much more. Moreover we can characterize theinvertible complex linear maps in terms of the real ones, namely

AutC(R2n) = GLn(C) = A ∈ GL2n(R) | AJ = JA

1Consider the set of all unit length quaternions S3 ⊂ R4, then we see that the subset ofall purely imaginary ones, meaning an equator of S3, is a 2-sphere.

126

Appendix

Example 8.1. Define a map

F : R4 → R2, (x, y, z, w) 7→ñ

x + zy + w

ô.

A complex structure for R2 is given by J2 from the observations above andfor R4

J4 :=ñ

J2 00 J2

ôdefines a complex structure.

Differentiating F yields

dF =

ñ1 0 1 00 1 0 1

ôhence F is clearly continuously differentiable in the real sense. Further wehave for v ∈ R4

J2dF(v) =ñ−v2 − v1v1 + v3

ôhence F is holomorphic dF(Jv) yields the same result. Fortunately it is

dF(J4v) =ñ−v2 − v1v1 + v3

ô,

thus F is holomorphic.

Note that changing the complex structure on a vector space corresponds toa change of basis. Therefore if we want to check if our map F is holomor-phic with respect to another J we would have change the matrix represen-tation of d f according to this change of basis as well. We will not need todo this, as the following theorem holds.

Theorem 8.2. Let V be a real even dimensional vector space and J1, J2 : V → Vtwo different complex structures. Then the following holds:

(i) J1 and J2 are conjugate to each other, i.e. there is a vector space isomor-phism T : V → V , such that J1 = T−1 J2T.

(ii) f : V → V is holomorphic with respect to J1 if and only if it is holomorphicwith respect to J2.

Proof. The proof will be a homework exercise.

127

On Algebraic Curves

In this part we want to consider the zero set of complex polynomials andtheir relation to Riemann surfaces. To deal with these we will recall theimplicit function theorem. Here we will use a slightly stronger versionthan the usual one for the Rk, but is assures that we can locally representthe zero set of some holomorphic function in two variables as the graph ofholomorphic map.

Theorem 8.3 (Implicit function theorem for holomorphic functions).Suppose that (z0, w0) is a point in X = (z, w) ∈ C2|p(z, w) = 0 and

∂p∂w

∣∣∣∣∣(z0,w0)

6= 0.

Then there is a disc D1 centered at z0 and D2 centered at w0 and a holomorphicmap φ : D1 → D2 with φ(z0) = w0, such that:

X ∩ (D1 × D2) = (z, φ(z))|z ∈ D1.

Affine Curves

Let p : C2 → C be a polynomial and X = (z, w) ∈ C2|P(z, w) = 0 thezero set.Suppose that at least one of the partial derivatives does not vanish. We willnow show that X is a Riemann surface.

Case 1): Suppose that we have (z0, w0) ∈ X with ∂p∂w 6= 0. Then we consider

the set Uα = (D1 × D2) ∩ X with (z0, w0) ∈ D1 × D2.We denote the projection map onto the first coordinate by

π1 : D1 × D2 → D1 : (x, y)→ x.

Then φα := π1|Uα is a coordinate chart of Uα.

128

Appendix

Case 2): Now suppose that we have (z1, w1) ∈ X with ∂p∂z 6= 0. Then we can

define the set Uβ = (B1 × B2) ∩ X, such that (z1, w1) ∈ B1 × B2.This time, we consider the projection onto the second component anddenote it by π2. We define the chart φβ := π2|Uβ

.

It remains to show that the defined carts are compatible.

It suffices to consider the case Uα ∩Uβ 6= ∅, because in the case of Uα ∩Uα′ 6= ∅ we just obtain the identity map as coordinate change.So let z ∈ Uα ∩Uβ be arbitrary. Then we observe:

zφ−1

α7−→ (z, f (z))φβ7−→ p(z)

This is clearly a holomorphic map since p was a polynomial! Thus X is aRiemann surface. But note that it is not necessarily compact.

Projective Curves

Recall: The complex projective space CPn is defined as

Cn+1 \ 0∼

wherev ∼ w :⇔ v = λw for λ ∈ C \ 0 .

Figure 8.1: One possible way to visualize about RP2. We can think of CPn

in a similar manner.

Let [z0, ..., zn] = [λz0, ..., λzn] ∈ CPn be arbitrary. We call these coordinateshomogeneous coordinates. Due to the fact that we consider equivalence classeswe can scale with our factor λ in such a way that that we can have 1 or 0 inour first component. This gives us the opportunity to transfer a problem inn variables in Cn into CPn, which can be useful in many applications.

129

Definition 8.4 (Homogeneous polynomial). A polynomial p is called ho-mogeneous polynomial of degree d if

p(z0, ..., zn) =∑

i0,...,in

ai0,...,in zi00 · ... · z

in

where∑n

j=0 ij = d and ik ∈N.

Let us now focus on the case that n = 2.

We consider a homogeneous polynomial p(z0, ..., z2) of degree d. We as-sume that z0 does not divide p. We define the set

X = [z0, z1, z2] ∈ CP2|p(z0, z1, z2) = 0.

Now we consider a slightly different polynomial p(w, z) = p(1, w, z). Wenote that p(w, z) defines an affine curve in CP2.If additionally we have

∂ p∂w

,∂ p∂v6= 0,

we can use our theory from above. We can conclude that the set

X0 = X ∩ [z] ∈ CP2|z0 6= 0

is a Riemann surface. If we put the ideas together, we obtain the followingtheorem:

Theorem 8.5. Suppose p(z0, z1, z2) is a homogeneous polynomial of degree d ≥1 and the only solution of the equation ∂p

∂z0= ∂p

∂z1= ∂p

∂z2= 0 is (0, 0, 0). Then

the solutions of the equation p = 0 in CP2 form a compact Riemann surface.

Proof. The proof is left as a homework exercise.

On Connections on Immersed Surfaces in R3

In the lecture we defined a connection as a substitute for a derivative onvector bundles, but apart from the trivial connection on the trivial bundle,which mimics the usual derivative, we lack of examples.

We will show a way of how to gain a canonical connection, the so called co-variant derivative on immersed surfaces f : U → R3 and convince ourselvesthat this really defines a connection for the case of M = S2 ⊂ R3.

130

Appendix

So consider the unit sphere S2 ⊂ R3. On R3, seen as a vector bundle, wehave the trivial connection denoted by d.A smooth map XS2 → R3 is a vector field on S2 if and only if¨

Xp, p∂= 0

for all p ∈ S2, where 〈., .〉 is the usual euclidean scalar product on R3.

We define a connection on S2 by

(∇XY)p := dpY(Xp)−¨dpY(Xp), p

∂p

which is nothing more than the projection of the usual derivative on R3 tothe respective tangent plane of S2.

Proposition 8.6. The map ∇ as defined above is a connection on S2.

Proof. The bilinearity is clear as both, d and 〈., .〉 are bilinear. Let f ∈ C∞(S2)and X, Y ∈ Γ(TS2) then

∇ f XY = dY( f X)− 〈dY( f X), p〉 p

= f X ·Y− f 〈X ·Y, p〉 p= f (dY(X)− 〈dY(X), p〉 p)= f∇XY

so ∇ is tensotial in X. Further

∇X( f Y) = d( f Y)X− 〈d( f Y)X, p〉 p= X · ( f Y)− 〈X · ( f Y), p〉 p= (X f )Y + f dY(X)− 〈(X f )Y, p〉 p− f 〈dY(X), p〉 p= (X f )Y + f∇XY

131

where we used that 〈(X f )Y, p〉 = 0 as Y is a vector field on S2. Thus ∇satisfies the product rule in the Y-component. The last thing to check isthat ∇ is well defined, that is that ∇XY really is a vector field on S2. Wehave

〈∇XY, p〉 = 〈dY(X)− 〈dY(X), p〉 p, p〉= 〈dY(X), p〉 − 〈dY(X), p〉〈p, p〉= 0

as 〈p, p〉 = 1 because p ∈ S2.

As already mentioned, the above construction yields a connection for anyimmersed surface in R3.

On Differential Forms

We start with the following basic observation:

Consider a smooth manifold M and some point p ∈ M. Identify a coor-dinate chart ϕ = (x1, . . . , xn) that is defined in a neighborhood of p withx = (x1, . . . , xn).

Then for a tangent vector u ∈ TpM we have

up =n∑

i=1ui(p) ∂

∂xi

∣∣∣p

and as the chart provides a frame-field in a neighborhood around p wehave

u =n∑

i=1ui

∂∂xi

.

For some function f ∈ C∞(M) we get

u f =n∑

i=1ui

∂ f∂xi

= d f (u).

132

Appendix

An important special case fo this is f = xj where xj is the j-th coordinateprojection. Then the upper forumula yields

uxj =n∑

i=1ui

∂xj∂xi

= uj = dxj(u)

and evaluating at p gives

(uxj)p =n∑

i=1ui(p) ∂xj

∂xi

∣∣∣p= dxj p(up)

thus as dxj p : TpM → R is linear and dxj p ∈ T∗p M we more generally getthat

d f p ∈ T∗p M, d f p =n∑

i=1

∂ f∂xi

(p)dxi p.

Use the frame field for the chart (U, x) to get sections of T∗U the so calledcotangent bundle.

Remark 8.7. Let V be a Hilbert space. By the Riesz-representation theoremfor each x∗ ∈ V∗ and y ∈ V there is x ∈ V such that

x∗y = 〈x, y〉 .

This leads to the definition of the so called musical isomorphisms which turna vector into a covector and a covector into a vector. This works as follows

] : T∗M→ TM, v∗ 7→ (v∗)] = v

[ : TM→ TM, v 7→ v[ = v∗

where the equality has to be understood in the sense of Riesz-represetnationabove.

In the lecture we defined

Λk(V) := ω : V × . . .×V → R | ω is k-linear and alternating .

If one has ω ∈ Ωk(M) and η ∈ Ω`(M), then is there some kind of ”multi-plication” that gives us a k + `-form from ω ane η?

Naturally one could try and use the usual tensor product defined by

(ω⊗ η)(X1, . . . , Xk, Y1, . . . , Y`) := ω(X1, . . . , Xk)η(Y1, . . . , Y`)

but the problem is, that in general

ω⊗ η /∈ Ωk+`(M).

133

Remedy is provided by the following alternation map which is defined asfollows:

Alt(ωp)(X1, . . . , Xk) :=1k!∑

σ∈Sk

sgn σ ωp(Xσ(1), . . . , Xσ(k)

which acts as an orthogonal projection in the space of tensors onto the spaceof alternating multilinear forms. Thus we can define the following

Definition 8.8 (Wedge product). Let ω ∈ Ωk(M) and η ∈ Ω`(M), then ak + `-form ω ∧ η ∈ Ωk+`(M) on M is defined by

ω ∧ η(X1, . . . , Xk+`) :=(k + `)!

k!`!Alt(ω⊗ η)(X1, . . . , Xk+`).

Remark 8.9. Λk can also be seen as a quotient on the tensor space with theequivalence relation that is given by

ω⊗ η ∼ ω⊗ η :⇔ Alt(ω⊗ η) = Alt(ω⊗ η).

Example 8.10.

dx ∧ dy = 2 Alt(dx⊗ dy) = dx⊗ dy− dy⊗ dx

Thus if we plug in vector fields X and Y we get

dx ∧ dy(X, Y) = dx(X)dy(Y)− dx(Y)dy(X) = X1Y2 − X2Y1

which is the usual formula for the derivative.

The following is majorly copied from the Differential Geometry 2 scriptwhere differential forms in general are usually introduced. Therefore mostof the things are in a much more general setup then we will use them,because as we are working with Riemann-surfaces there will only be 1-, or2-forms.

Bundle-Valued Differential Forms

Definition 8.11 (Bundle-valued differential forms). Let E → M be avector bundle. Then for ` > 0 an E-valued `-form ω is a section of thebundle Λ`(M, E). We write Ω`(M, E) := Γ

(Λ`(M, E)

). Further, define

Λ0(M, E) := E. Consequently, Ω0(M, E) := Γ(E).

134

Appendix

Definition 8.12 (Exterior derivative). Let E → M be a vector bundle withconnection ∇. For ` ≥ 0, define the exterior derivative

d∇ : Ω`(M, E)→ Ω`+1(M, E)

for vectors X0, . . . , X` ∈ Γ(TM) as follows:

d∇ω(X0, . . . , X`) :=∑

i(−1)i∇Xi(ω(X0, . . . , Xi, . . . , X`))

+∑i<j

(−1)i+jω([Xi, Xj], X0, . . . , Xi, . . . , Xj, . . . , X`)

`-forms:Let M ⊂ Rn be open and consider again

E = M×R.

Then for i1, . . . , i` define dxi1 ∧ · · · ∧ dxi` ∈ Ω`(M) by

dxi1 ∧ · · · ∧ dxi`(X1, . . . , X`) := det

Üdxi1(X1) · · · dxi1(X`)

... . . . ...dxi`(X1) · · · dxi`(X`)

ê.

Note: If iα = iβ for α 6= β, then dxi1 ∧ · · · ∧ dxi` = 0 and if

σ : 1, . . . , ` → 1, . . . , `

is a permutation, we have

dxiσ1∧ · · · ∧ dxiσ`

= sign σ dxi1 ∧ · · · ∧ dxi` .

Theorem 8.13. Let U ⊂ Rn be open. The `-forms dxi1 ∧ · · · ∧ dxi` for 1 ≤i1 < · · · < i` ≤ n are a frame field for Λ`(U), i.e. each ω ∈ Ω`(U) can beuniquely written as

ω =∑

1≤i1<···<i`≤nai1···i` dxi1 ∧ · · · ∧ dxi`

with ai1···i` ∈ C ∞(U). In fact,

ai1···i` = ωÄ ∂

∂xi1, . . . ,

∂

∂xi`

ä.

135

Theorem 8.14. Let U ⊂ Rn be open and ω =∑

1≤i1<···<i`≤nai1···i` dxi1 ∧ · · · ∧

dxi` ∈ Ω`(U), then

dω =∑

1≤i1<···<i`≤n

n∑i=1

∂ai1···i`∂xi

dxi ∧ dxi1 ∧ · · · ∧ dxi` .

Proof. By Theorem 8.13 it is enough to show that for all 1 ≤ j0 < · · · < j` ≤n we have

dωÄ ∂

∂xj0, . . . ,

∂

∂xj`

ä=

∑1≤i1<···<i`≤n

n∑i=1

∂ai1···i`∂xi

dxi ∧ dxi1 ∧ · · · ∧ dxi`

Ä ∂

∂xj0, . . . ,

∂

∂xj`

ä=∑k=0

∂aj0··· jk···j`∂xjk

dxjk ∧ dxj0 · · · ∧‘dxjk · · · ∧ dxj`

Ä ∂

∂xj0, . . . ,

∂

∂xj`

ä=∑k=0

(−1)k∂aj0··· jk···j`

∂xjk.

But we also get this sum if we apply the definition and use that [ ∂∂xk

, ∂∂xm

] =0.

Example 8.15. Let M = U ⊂ R3 be open. Then every σ ∈ Ω2(M) can beuniquely written as

σ = a1dx2 ∧ dx3 + a2dx3 ∧ dx1 + a3dx1 ∧ dx2.

Let σ = dω with ω = v1dx1 + v2dx2 + v3dx3. Then

dωÄ ∂

∂xi,

∂

∂xj

ä=

∂

∂xiωÄ ∂

∂xj

ä− ∂

∂xjωÄ ∂

∂xi

ä=

∂vj

∂xi− ∂vi

∂xj.

Thus we get that a = curl(v).

The proofs of Theorem 8.13 and Theorem 8.14 directly carry over to bundle-valued forms.

Theorem 8.16. Let U ⊂ Rn be open and E → U be a vector bundle withconnection ∇. Then ω ∈ Ω`(U, E) can be uniquely written as

ω =∑

1≤i1<···<i`≤nψi1···i` dxi1 ∧ · · · ∧ dxi` , ψi1···i` ∈ Γ(E).

Moreover,

d∇ω =∑

1≤i1<···<i`≤n

n∑i=1

Ä∇ ∂

∂xiψi1···i`

ädxi ∧ dxi1 ∧ · · · ∧ dxi` .

136

Appendix

Pullback

Motivation: A geodesic in M is a curve γ without acceleration, i.e. γ′′ =(γ′)′ = 0.

But what a map is γ′? What is the second prime?

We know that γ′(t) ∈ Tγ(t)M. Modify γ′ slightly by

γ′(t) = (t, γ′(t)) meaning that γ′ ∈ Γ(γ∗TM).

Right now γ∗TM is just a vector bundle over (−ε, ε). If we had aconnection ∇ then we can define

γ′′ = ∇ ∂∂t

γ′.

Definition 8.17 (Pullback of forms). Let f : M → M be smooth and ω ∈Ωk(M, E). Then define

f ∗ω ∈ Ωk(M, f ∗E)

by( f ∗ω)(X1, . . . , Xk) := (p, ω(d f (X1), . . . , d f (Xk)))

for all p ∈ M, X1, . . . , Xk ∈ TpM.For ψ ∈ Ω0(M, E) we have f ∗ψ = (Id, ψ f ).

For ordinary k-forms ω ∈ Ωk(M) ∼= Ωk(M, M×R):

( f ∗ω)(X1, . . . , Xk) = ω(d f (X1), . . . , d f (Xk)).

Let E→ M be a vector bundle with connection ∇, f : M→ M.

Theorem 8.18. There is a unique connection

∇ =: f ∗∇

on f ∗E such that for all ψ ∈ Γ(E), X ∈ TpM we have ∇X( f ∗ψ) =Äp, ∇d f (X)ψ

ä.

In other words∇( f ∗ψ) = ( f ∗∇)( f ∗ψ) = f ∗(∇ψ).

137

Proof. For uniqueness we choose a local frame field ϕ1, . . . , ϕk around f (p)defined on V ⊂ N and an open neighborhood U ⊂ M of p such thatf (U) ⊂ V.Then for any ψ ∈ Γ( ( f ∗E)|U) there are g1, . . . , gk ∈ C ∞(U) such that ψ =∑

jgj f ∗ϕj. If a connection ∇ on f ∗E has the desired property then, for

X ∈ TpM,

∇Xψ =∑

j

Ä(Xgj) f ∗ϕj + gj∇X( f ∗ϕj)

ä=∑

j

Ä(Xgj) f ∗ϕj + gj(p, ∇d f (X)ϕj)

ä=∑

j

Ä(Xgj) f ∗ϕj + gj

∑k(p, ωjk(X)ϕk)

ä= (p,

∑j

Ä(Xgj)ϕj f + gj

∑k

ωjk(X)ϕk f )ä,

where∇d f (X)ϕj =

∑k

ωjk(X)ϕk f ,

with ωjk ∈ Ω1(U). For existence check that this formula defines a connec-tion.

Theorem 8.19. Let ω ∈ Ωk(M, U), η ∈ Ω`(M, V) and ∗ ∈ Γ(U∗⊗V∗⊗W),then

f ∗(ω ∧ η) = f ∗ω ∧ f ∗η.

Proof. Trivial.

Theorem 8.20. Let E be a vector bundle with connection ∇ over M, f : M →M, ω ∈ Ωk(M, E), then

d f ∗∇( f ∗ω) = f ∗(d∇ω).

Proof. Without loss of generality we can assume that M ⊂ Rn is open andthat ω is of the form

ω =∑

1≤i1<···<ik≤nψi1···ik dxi1 ∧ · · · ∧ dxik .

Then

f ∗ω =∑

1≤i1<···<ik≤n( f ∗ψi1···ik) f ∗dxi1 ∧ · · · ∧ f ∗dxik ,

d∇ω =∑

1≤i1<···<ik≤n∇ψi1···ik ∧ dxi1 ∧ · · · ∧ dxik .

138

Appendix

Hence

f ∗d∇ω =∑

1≤i1<···<ik≤nf ∗(∇ψi1···ik) ∧ f ∗dxi1 ∧ · · · ∧ f ∗dxik

=∑

1≤i1<···<ik≤n( f ∗∇ f ∗ψi1···ik) ∧ dxi1 d f ∧ · · · ∧ dxik d f

=∑

1≤i1<···<ik≤n(d f ∗∇ f ∗ψi1···ik) ∧ d(xi1 f ) ∧ · · · ∧ d(xik f )

= d f ∗∇( f ∗ω).

Exercise 8.21.Consider the polar coordinate map f : (r, θ) ∈ R2 | r > 0 → R2 given byf (r, θ) := (r cos θ, r sin θ) = (x, y). Show that

f ∗(x dx + y dy) = r dr and f ∗(x dy− y dx) = r2 dθ.

Theorem 8.22 (Pullback metric). Let E → M be a Euclidean vector bundlewith bundle metric g and f : M → M. Then on f ∗E there is a unique metricf ∗g such that

( f ∗g)( f ∗ψ, f ∗φ) = f ∗g(ψ, φ)

and f ∗g is parallel with respect to the pullback connection f ∗∇.

Exercise 8.23.Prove Theorem 8.22.

On Stokes Theorem

We want to give a proof of Stokes theorem in n-dimensions.

Proof. The proof will consist of three steps. At first we consider the caseM = Hn, then for ω ∈ Ωn−1

0 (Hn) there exists some R > 0 such that

supp ω ⊂ [−R, 0]× [−R, R]n−1.

We can writeω =

∑i

ωi dx1 ∧ . . . ∧‘dxi ∧ . . . ∧ dxn

and taking the cartan-derivative gives

dω =∑

i(−1)i−1 ∂ωi

∂xidx1 ∧ . . . ∧ dxn.

139

Evaluating the lefthand side of the equation gives∫Hn

dω =n∑

i=1

∫ R

−R. . .∫ R

−R

∫ 0

−R(−1)i−1 ∂ωi

∂xidx1 . . . dxn

Using Fubini, for i = 2, . . . , n we integrate first with respect to i whichyields ∫ R

−R

∂ωi

∂xi= ωi|R−R = 0

as ω is compactly supported. So with this∫Hn

dω =∫ R

−R. . .∫ R

−Rω(0, x2, . . . , xn)dx2 . . . dxn.

On the righthand side we get∫∂Hn

ω =n∑

i=1

∫ R

−R. . .∫ R

−Rωi(0, x2, . . . , xn)dx1 . . .‘dxi . . . dxn

As dx1|∂M = 0 since x1 ≡ 0 at the boundary we get that the lefthand- andthe righthand-side are equal.As the second step let M be a manifold with boundary and supp ω ⊂ Ufor a chart (U, ϕ). Then∫

Mdω =

∫ϕ(U)

(ϕ−1)∗dω =∫

ϕ(U)d((ϕ−1)∗ω) =

∫∂Hn∩ϕ(U)

(ϕ−1)∗ω =∫

∂Mω.

As third step let supp ω ⊂ U1 ∪ . . . ∪Um, then choose a partition of unitysubordinate to the cover (Ui). We get∫

∂Mω =

m∑i=1

∫∂M

$iω =m∑

i=1

∫M

d($iω) =m∑

i=1

∫M

d$i ∧ω + $idω

=∫

Md

Ñm∑

i=1$i

é∧ω +

∫M

m∑i=1

$idω =∫

Mdω.

On the Definition of Vector Bundles

Recall the definition of a vector bundle.

Definition 8.24 (Vector bundle). A smooth vector bundle of rank k is a triple(E, M, π) where E and M are manifolds and π : E→ M is smooth such that

(i) The fiber Ep := π−1 (p) has the structure of a k-dimensional realvector space.

140

Appendix

(ii) There is U ⊂ M open, p ∈ U and a diffeomorphism φ : π−1(U) →U ×Rk such that π1 φ = π|U where π1 is the projection on the firstcomponent, i.e. for each q ∈ U the map φq : Eq → Rk is a vector spaceisomorphism defined by

Äq, φq(ψ)

ä= φ(ψ).

In the literature (e.g. the script on Riemann surfaces of Prof. Bobenko) onemay stumble across a different definition of a vector bundle. We want toshow that this in fact is an equivalent definition to ours. We will formulateit as a theorem.

Theorem 8.25. Let E be a rank k vector bundle. Then there are unique holomor-phic mappings gij : Ui ∩Uj → GLk(C) such that for the transition functionsφij = φj φ−1

i : (Ui ∩Uj) × Ck → (Ui ∩Uj) × Ck it holds that φij(p, e) =(p, gij(p)e). Furthermore the cocycle-realtion holds, i.e. gijgjk = gik.

The proof of this theorem should be clear by now. The more interestingimplication is the converse of the theorem yielding equivalence of the defi-nitions.

Theorem 8.26. Suppose M is a Riemann surface, U = (Ui)i∈I an open cover-ing of M and (gij)i,j∈I endomorphism-valued functions subordinate to U withgij(p) ∈ GLn(C) for all p ∈ Ui ∩Uj, which satisfy the cocycle-relation. Thenthere exists some holomorphic vector bundle (E, M, π) of rank n and a holomor-phic atlas

¶φi : EUi → Ui ×Cn | i ∈ I

©of E whose transition functions are the

given gij’s

Proof. Let E′ := M× C2 × I. The plan is to define an equivalence relation∼ on E′ such that E′/ ∼ is the desired vector bundle. Equip E′ with theinduced topology by M, Cn and I with the discrete topology. We introducethe equivalence relation

(p, e, i) ∼ (p′, e′, j) :⇔ p = p′ and e′ = gij(p)e.

This is an equivalence relation, because the transitivity follows from thecocycle-relation and reflexivity and symmetry are clear.Define E := E′/ ∼ with the quotient topology, then E is Hausdorff as wehave that E′ and M are Hausdorff and

E′

M

E

π′1

ϕ

π

141

holds with all continuous maps. The equivalence relation ∼ is compatiblewith π′1 which implies that π1 is continuos. Further π−1

1 (p) has vectorspace structure as

π−11 (p) = p ×Cn × i ∼= Cn

By the map id× gij × i 7→ j we have an isomorphsim

p ×Cn × i ∼= Cn.

We still need to check if E is locally trivial. This is given as

π−11 (Ui) = (ϕ π′1)(Ui) = ϕ(Ui ×Cn × I) = EU−i.

ϕ|Ui×Cn×i is a homeomorphism onto EUi . Now define

φi : EU−i → Ui ×Cn

as the inverse of ϕ, then by construction

φi(p, e) = (p, gij(p)e).

On The Tangent Bundle of CPn

Now our aim is to show that

T[x]CPn = HomÇ

L, Cn+1 \ 0L

åwhere L = π−1([x]). Further note that the quotient space in this exampleis the usual one defined by the equivalence relation x ∼ y :⇔ x− y ∈ L.So let X ∈ T[x]CPn be arbitrary. Then we know from the definition of atangent vector that there is a curve γ : (−ε, ε)→ CPn such that

X f = ( f γ)′(0) for all f ∈ C∞(CPn)

Now we consider a lift of γ this is some curve γ : (−ε, ε)→ Cn+1\0 suchthat π γ = γ. We can illustrate this in the following commuting diagram

(−ε, ε)

Cn+1\0

CPn

γ

γ

π

142

Appendix

If we fix some p ∈ L we can define the tangent vector γ(0) ∈ Cn+1\0 atp. Then we obtain

d π(γ) = X

This definition does not depend on the choice of the lift.To see this let γ, γ be two lifts of γ with γ(0) = p = γ(0). Then there issome functionf : (−ε, ε)→ C\0 with f (0) = 1. This yields

ddt

∣∣∣∣∣t=0

γ(t) = f ′(0) · p︸︷︷︸∈L

+ f (0)︸︷︷︸=1

· ˙γ(0)

Thus we can define for a curve γ : (−ε, ε) → Cn+1\0 with γ(0) = p thehomomorphism

X : L→ Cn+1 \ 0L : p 7→[

ddt

∣∣∣∣∣t=0

γ

]

At first this might seem a bit weird but with a closer look one can see thatthese definitions of X coincide and this yields the claim.

On the Curvature 2-form

After deriving the curvature 2-form Ω of a Riemannian curvature tensorone may ask the question of how to compute it explicitly.

Let L be a complex line bundle over M. Then we know that we have localtrivializations (Uα, φα)α∈I with φα : π−1(Uα)→ Uα×C. How do we expressa connection ∇ in these terms?

Choose a frame sα = φ−1α (p, 1). Then every section S ∈ Γ(L) can locally be

expressed ass|Uα = fα · sα

for fα : Uα → C. Further, on Uα ∩Uβ we have that

sα = gαβsβ.

We can express a connection ∇ in terms of the trivial connection d and atensor A by

∇ = d + A.

Locally ∇ can be expressed as

(∇S)∣∣∣Uα

= ∇( fαsα) = d( fαsα) + fα∇sα,

143

hence∇sα := Aαsα ∈ Ω1.

Note that on the one hand

∇sα = Aαsα = Aαgαβsβ

and on the other hand

∇sα = ∇(gαβsβ) = d(gαβsβ) + gαβ∇sβ = d(gαβsβ) + gαβ Aβsβ.

Hence the compatibility condition on Aα and Aβ is given by

Aβ = g−1αβ Aαgαβ − g−1

αβ dgαβ.

Further, from the homework we know that

R∇ = d∇d∇∣∣∣Γ(L)

.

This gives, for ψ ∈ Γ(L) with ψ = f · s on Uα the local expression

R∇ψ = d(d∇ψ) = d(∇ψ) = d(∇( f s)) = d((d f )s + f∇s) = d((d f )s + f (As))= d((d f + f A)s) = d(d f + f A)s− (d f + f A) ∧ As= (dd f︸︷︷︸

=0

+d f ∧ A + f dA)s− (d f + f A) ∧ ∇s︸︷︷︸=As

= ( f dA + f A ∧ A) s= (dA + A ∧ A) f s= (dA + A ∧ A)ψ

Since L is a line bundle A ∧ A = 0 hence

Ωα = dAα.

We ca now do the same thing on Uβ and check that Ω is well defined bychecking that Ωα = Ωβ on Uα ∩Uβ.

Ωβ = dAβ = d(

aα − g−1αβ dgαβ

)= dAα − (dg−1

αβ ) ∧ dgαβ

= dAα + (g−1αβ dgαβg−1

αβ ) ∧ dgαβ︸︷︷︸=...dgαβ∧dgalphaβ=0

= dAα

= Ωα

An interesting observation we can make on the derivation of the formuladA = Ω is that R∇ = dA + A ∧ A. There is an even more general formulafrom which this statement follows directly.

144

Appendix

Lemma 8.27. Let ∇ = d + A be a connection on a vector bundle E over M.Then

R∇ = Rd + dA + A ∧ A.

Proof. For X, Y ∈ Γ(TM) and ψ ∈ Γ(E) we get

R∇(X, Y)ψ = ∇X∇Yψ−∇Y∇Xψ−∇[X,Y]ψ

= ∇X (dYψ + AYψ)−∇Y (dXψ + AXψ)− d[X,Y]ψ− A[X,Y]ψ

= dXdYψ + AXdYψ + dX(AYψ) + AX AYψ

− dYdXψ + AYdXψ + dY(AXψ) + AY AXψ

− d[X,Y]ψ− A[X,Y]ψ

= Rd(x, Y)ψ + dA(X, Y)ψ + A ∧ A(X, Y)ψ

Applying this theorem to the above case yields Rd = 0 as C ∼= R2 hasno curvature with respect to the trivial connection and A ∧ A = 0 as it iscomplex one dimensional. This immediately yields R∇ = dA.

On the curvature form of the sphere with radius R

In definition 6.19 we have seen the definition of the curvature 2-form. Nowit is our aim to use this definition and calculate this form for the sphere.As in the last section one might think that it is a constructive way to writeour Levi-Civita connection ∇ = d + A where A is a endomorphism valuedform. Here we have for X, Y ∈ Γ(S2)

(AYX)p = −〈dY(X), p〉p

We have seen that Ω = dA, but in fact this is for our concrete example nota good way to go.For a fixed radius R we consider spherical coordinates

f (ϕ, θ) =

ÖR cos(ϕ) cos(ϑ)R sin(ϕ) cos(ϑ)

R sin(ϑ)

èwhere ϕ ∈ [0, 2π), ϑ ∈

Ä−π

2 , π2

äand N = 1

R f . We obtain for the partialderivatives

∂ f∂ϕ

= R

Ö− sin(ϕ) cos(ϑ)cos(ϕ) cos(ϑ)

0

è,

∂ f∂ϑ

= R

Ö− cos(ϕ) sin(ϑ)− sin(ϕ) sin(ϑ)

cos(ϑ)

è145

We choose as a frame

e1 =

Ö− sin(ϕ)cos(ϕ)

0

èand e2 =


cos(ϑ)

è.

Hence we can write

e1 =1

R cos(ϑ)∂

∂ϕ, e2 =

1R

∂

∂ϑ

for our frame. This yields for the dual frame

e[1 = R cos(ϑ)dϕ, e[2 = R · dϑ

In particular

∇e1e1 = d(e1)e1 − 〈d(e1)e1, N〉N

=1

R cos(ϑ)∂

∂ϕ

Ö− sin(ϕ)cos(ϕ)

0

è− 1

R cos(ϑ)

±Ö− sin(ϕ)cos(ϕ)

0

è,

Öcos(ϕ) cos(ϑ)sin(ϕ) cos(ϑ)

sin(ϑ)

èªN

=1

R cos(ϑ)

ÖÖ− sin(ϕ)cos(ϕ)

0

è− cos(ϑ)N

è=

1R cos(ϑ)

Öcos(ϕ)(cos2(ϑ)− 1)sin(ϕ)(cos2(ϑ)− 1)

cos(ϑ) sin(ϑ)

è=

sin(ϑ)cos(ϑ)


cos(ϑ)

è= − tan(ϑ)

Re2

∇e1e2 =1

R cos(ϑ)∂

∂ϕ


cos(ϑ)

è− sin(ϑ)

R cos(ϑ)

±Ö− sin(ϕ)cos(ϕ)

0

è, N

ªN

=− tan(ϑ)

Re2

146

Appendix

∇e2e2 =1R

∂

∂ϑ


cos(ϑ)

è︸︷︷︸

− 1R N

− 〈de2(e2), N〉︸︷︷︸− 1

R

N = 0

∇e2e1 = 0

This leads us to the definition of the so called Christoffel symbols. We write

∇ej ei = Γkijek

where we use the Einstein sum convention. Thus the Christoffel-symbolsare just special scalar valued functions. We could write equivalently

∇ej ei = ωji(ei)⊗ ej,

where ω is a matrix valued connection 1-form.This yields

ω =

Ü0

tan(ϑ)R

e[1

− tan(ϑ)R

e[2 0

ê=

Ç0 sin(ϑ)

− sin(ϑ) 0

ådϕ

We can conclude for our curvature form

Ω = dω =

Ç0 cos(ϑ)

− cos(ϑ) 0

ådϑ ∧ dϕ = J cos(ϑ)dϑ ∧ dϕ

Note that this is consistent with our observation in theorem 6.22, where weshowed that the curvature form is purely imaginary. If we interpret ourtangent space as a one-dimensional line bundle we see that J ”becomes” i.We can use this to calculate the integral of the Gaußian curvature over thesphere. We have

K = 〈R(e1, e2)e2, e1〉 = − cos(ϑ)dϑ ∧ dϕ(e1, e2)

= − cos(ϑ)dϑ ∧ dϕ

Ç1

R cos(ϑ)∂

∂ϕ,

1R

∂

∂ϑ

å=

1R2

Thus1

2πi

∫RS2

Ω =1

2π

∫RS2

KdA =1

2πR2

∫RS2

1 dA = 2 = χS2

We can draw an important consequence.

Theorem 8.28. (Hairy ball theorem)Let X ∈ ΓTS2. Then there is some p ∈ S2 with Xp = 0

147

Proof. Assume that there is a vector field X ∈ ΓTS2 with no zeros.The Poincare-Hopf-Index theorem yields

0 =∑

iindpi X =

12πi

∫S2

Ω =1

2π

∫S2

K dA = 2

and therefore a contradiction.

On Homology

In the Riemann-Roch theorem it is our aim to calculate the Betti-numbers.These are defined as dimensions of special quotient spaces. To get a betterunderstanding to these we will now discuss the basic ideas of homologyand cohomology.

In our setting let M be a smooth, oriented, connected and compact man-ifold. Note that many of the concepts of homology and cohomology alsowork for more general manifolds. But we aim for the Riemann-Roch The-orem; thus these assumptions are necessary Let (ni)i∈I ∈ R be real coeffi-cients. Then we define

• C0 as the set of 0-chains p =∑

ini pi.This can be understood as the

formal sum of points pi ∈ M.

• C1 as the set of 1-chains γ =∑

iniγi. We can think of it as the formal

sum of curvesγi : [ai, bi]→ M

• C2 as the set of 2-chains D =∑

iniDi. Here D is a formal sum of

singular triangles on M

Note that a singular triangle is a continuous map

ϕ : (t1, t2) ∈ R2 : t1, t2 ≥ 0, t1 + t2 = 1︸︷︷︸:=∆

→ M.

Here in our special setting we even see that our k-chains form vector spacesover R. For curves we could think of the inverse element under formaladdition as the element that reverses the orientation.Now we can define a boundary operator on our chains.So let D be a singular triangle, γ be a curve and p a fixed point in M. Then

• ∂2 : C2 → C1 : D = (P1, P2, P3) 7→ (P1, P2) + (P2, P3) + (P3, P1)

148

Appendix

• ∂1 : C1 → C0 : γ = (P1, P2) 7→ P2 − P1

We define important subgoups (here even subspaces) of our chains:

• A k-chain γ is called a cycle, if ∂kγ = 0.We set Zk(M) = c ∈ Ck(M) | ∂c = 0 = ker(∂k)

• A k-chain γ is called a boundary, if there is a k + 1-chain D such that∂k+1D = γ.We define Bk(M) = c is boundary = im(∂k+1)

It is trivial to see that ∂2 = 0. Thus Bk(M) ⊂ Zk(M). This leads to thedefinition

Hk(M) = Zk(M)Bk(M) =ker ∂kim ∂k+1

of the k-th homology.

Remark 8.29. This type of homology is also known as singular homology.

Definition 8.30 (homologous). Let γ1,γ2 ∈ H1(M). They are called ho-mologous if there exists a D such that

γ1 − γ2 = ∂D.

By the definition of Hk(M) we see that homologous curves are in Hk(M)considered as the same thing. This leads to the natural question whichelement lie in Hk(M).

It is easy to believe that on the 2-sphere all boundary curves of faces arehomologous to each other. Thus one can believe that dim Hk(S

2) = 0.But on the torus we can see that the curves γ1 and γ2 are not homologous.Nevertheless it is easy to believe that every other boundary curve of a faceis homologous to a formal linear combination of γ1 and γ2. Our intuitionis that dim(Hk(T2)) = 2.In order to proof this, remember that the k-th cohomology is defined as

Hk(M) = ker (dk)im(dk−1).

We will now make use of the theorem of de Rham in the special case thatHk(M) and Hk(M) are vector spaces.

149

Theorem 8.31. (de Rham)The map

I : Hk(M)→ (Hk(M))∗ : ω 7→Å

c 7→∫

cωã

is an isomorphism of vector spaces.

We know from the Riemann-Roch-Theorem that the dimension of the k-thcohomology is always finite. Thus by the De-Rham theorem we obtain thatdim(Hk(M)) = βk(M).We calculated last time that deg(TS2) = χ(S2) = 2. A similar calculationshows thatdeg(T(T2)) = χ(T2) = 0. The Riemann-Roch-theorem yields

• dim(H1(S2)) = β1(S2) = 2− χ(S2) = 0

• dim(H1(T2)) = β1(T2) = 2− χ(T2) = 2

This shows in fact that our considerations above on the homology were infact right.

On a Discrete Gauss-Bonnet Theorem

From now on we will only consider immersed simplicial surfaces M withvertex set V, edges E and faces F. We first of all need to clarify what thecurvature of such a simplicial surface is. We define the unit normal Nijk ofthe triangle ijk by

N : F → S2, ijk 7→ Nijk

where Nijk is the unit normal of the oriented plane in R3 which is uniquelydetermined by three vertices of the triangle and their orientation. Analo-gously the interior angle θ

jki at the corner i of ijk is defined as

θ : F → R, ijk 7→ θjki

where again θjki is the corresponding angle of the triangle as indicated in

the sketch.

150

Appendix

Remark 8.32. Consider a simplex ijk ∈ F, then its unit normal, as well asthe interior angles, are determined by the positions of the simplex ijk in R3.We can explicitly compute Nijk and θ

jki as functions of i, j, k with

Nijk =(k− i)× (j− i)|(k− i)× (j− i)|

andθ

jki = arccos

Æk− i|k− i| ,

j− i|j− i|

∏.

Definition 8.33 (angle defect). The angle defect of an interior vertex i ∈ M

is given by the sumΩi := 2π −

∑ijk∈F

θjki .

The area of a spherical triangle with interior angles α1, α2, α3 is given by

A = α1 + α2 + α3 − π

thus for a spherical polygon with n ∈N vertices and interior angles α1, . . . , αn,we yield

A = (2− n)π +n∑

i=1αi = 2π −

n∑i=1

(π − αi)

by triangulating the polygon and consecutive application of the formulafor a spherical triangle. With some geometric observations we see that theexterior angles of the spherical polygon N(St(i)) given by the Gauss-imageof a vertex star St(i) are given by θ

jki , for ijk ∈ F. Thus we yield that

AN(St(i)) = 2π −∑

ijk∈F

Åπ −

Åπ − θ

jki

ãã= 2π −

∑ijk∈F

θjki

meaning that the angle defect corresponds to the area of the spherical tri-angle that is determined by the Gauss-map of the unit normals to each facethat contains the vertex i.With the link to the smooth theory where a well known interpretation of theGaussian curvature is as the ratio of the area enclosed by the Gauss map,to surface area of a neighbourhood around a vertex i ∈ V, the followingdefinition makes perfect sense.

Definition 8.34 (discrete Gaussian curvature). The Gaussian curvatureof an interior vertex i ∈ M is given by the angle defect and set to zero for

151

boundary vertices.

The total Gaussian curvature of a simplicial surface M is therefore given by

K(M) =∑i∈M

Ωi.

How well the characteristics of curvature are satisfied by the discretiza-tion of Gaussian curvature is also seen by the fact that the broadly knownGauss-Bonnet theorem also translates to our discrete setting. We use thatthe Euler characteristic of a simplicial surface M is given by

χ(M) = |V| − |E|+ |F|.

Theorem 8.35 (Gauss-Bonnet for simplicial surfaces). Let f : M → R3 bea simplicial surface without boundary, then∑

i∈VΩi = 2πχ(M).

Proof. By the Definition of Ωi, we have

∑i∈V

Ωi =∑i∈V

Ñ2π −

∑ijk∈F

θjki

é=∑i∈V

2π −∑i∈V

Ñ∑ijk∈F

θjki

é= 2π|V| −

∑i∈V

Ñ∑ijk∈F

θjki

éSince the interior angles of every face ijk ∈ F add up to π we yield

∑i∈V

Ñ∑ijk∈F

θjki

é=

∑ijk∈F

(3− 2)π = π (2|E| − 2|F|)

where we have used that every edge is shared by 2 faces. Thus∑i∈V

Ωi = 2π|V| − π (2|E| − 2|F|) = 2πχ(M)

Remark 8.36. The theorem above extends also to simplicial surfaces with aboundary ∂M. In a similar manner it is possible to proof that∑

i∈MΩi +

∑i∈∂M

Γi = 2πχ(M)

152

Appendix

where Γi denotes the discrete geodesic curvature of a boundary vertex i ∈∂M defined by

Γi := π −∑

ijk∈Fθ

jki .

153

complex analysis ii - tu berlin · ii. 1.introduction the ﬁrst formal deﬁnition of a manifold...

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