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On the Maslov Index

July 25, 2011

Gouri Shankar SealIntegrated MS 5th yearIISER kolkataMathematics Department

University Lyon 1Institute Camille JordanSummer Project Report

Submitted under supervision ofDr. Jean Yves Welschinger

Abstract

In this article we study briefly the theory of Maslov index for a pathof lagrangian submanifolds following Robbin Salamon. In the process themost important properties of Maslov index will be discussed. We will alsobriefly investigate Arnold’s definition of Maslov index and define it fora compact Riemann surface with boundary. After stating the Riemann-Roch theorem for compact Riemann surface with boundary we will use itto calculate the dimension of the space of pseudoholomorphic curves inan almost complex manifold of real dimension four.

Contents

1 The Lagrangian grassmanian from the ordinary grassmannian 21.1 Charts of Gk(n): . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Tangent space to a grassmannian . . . . . . . . . . . . . . . . . . 31.3 The Lagrangian grassmanian . . . . . . . . . . . . . . . . . . . . 41.4 The submanifolds Λk(L0) . . . . . . . . . . . . . . . . . . . . . . 6

2 A brief review of the theory of symmetric bilinear forms 7

3 Maslov index for lagrangian paths 93.1 Maslov Index for lagrangian pairs . . . . . . . . . . . . . . . . . . 123.2 Maslov Index for Symplectic paths . . . . . . . . . . . . . . . . . 15

4 Arnold’s interpretation and the boundary Maslov index 204.1 Boundary Maslov index . . . . . . . . . . . . . . . . . . . . . . . 21

5 Cauchy Riemann operators and application of the Riemann-Roch formula 245.1 Introduction to Cauchy-Riemann operators . . . . . . . . . . . . 245.2 Real linear Cauchy-Riemann operators . . . . . . . . . . . . . . . 255.3 Dimension of the space of pseudoholomorphic curves in an almost

complex manifold . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1

1 THE LAGRANGIANGRASSMANIAN FROMTHEORDINARYGRASSMANNIAN2

1 The Lagrangian grassmanian from the ordinary grassmannian

In this section we will study in detail the structure and properties of thegrassmannian and use it to explore the geometry of the lagrangian grassman-nian.Definition 1.1: Let n, k be fixed integers with n ≥ 0 and 0 ≤ k ≤ n we willdenote by

Gk(n) = V ⊂ Rn|dim(V ) = k

called the grassmannian of k dimensional subspaces of Rn. Decompose Rn =W0 ⊕W1 where dim(W0) = k then for every linear operator

T : W0 →W1

define the graph of T by Gr(T ) = v + Tv|v ∈W0 is an element of Gk(n).Moreover W ∈ Gk(n) is of the form Gr(T) iff it is transversal to W1 i.e., W ∈Gk(n,W1) where we have defined the open subspaces Gk(n,W1) as

Gk(n,W1) = W ∈ Gk(n)|W ∩W1 = φ

Hence we can define the bijection

φW0,W1: Gk(n,W1)→ Lin(W0,W1)

by setting φW0,W1(W ) = T where W = Gr(T )

Remark:

• It is easy to visualize all the lagrangian subspaces(isomorphic to RP 1) indimension n = 1 with the ambient space being R2. Set W0 = Rn × 0 andW1 = 0× Rn then the graph of a linear map from the horizontal to the

vertical is a line passing through the origin which is precisely thelagrangian grassmannian

Fig 1: The Lagrangian grassmannian is isomorphic to RP 1 in dimension n = 1

• More concretely if π0 and π1 denotes the projections along W0 and W1

in the decomposition Rn = W0 ⊕W1 then the operator T = φW0,W1is

uniquely represented as T = (π1)|W (π0)−1|W .

We simply make this small observationn that W ∩W1 = φ is equivalentto the condition that π0|W is an isomorphism


Definition 1.2: Given subspaces W0,W′

0 ⊂ Rn and given a common com-plementary subspace W1 ⊂ Rn i.e., Rn = W0 ⊕W1 = W

′

0 ⊕W1 we have anisomorphism

η = ηW1

W0,W′0

: W0 →W′

0

obtained by restriction to W0 of the projection onto W0 relative to Rn

Rn/W1

qW0

qW′0

W0

ηW1

W0,W′0−→ W

′

0

where q : Rn → Rn/(W1) is the quotient map.

1.1 Charts of Gk(n):

Consider charts φW0,W1 and φW ′0 ,W1in Gk(n) with k = dim(W0) = dim(W

′

0)

then define the transition functionφW0,W1

φ−1

W′0 ,W1

(T ) = (π′

1|W0+ T ) ηW1

W0,W′0

where π′

1 : W′

0 ⊕W1 →W1. We therefore have the following proposition

Proposition 1.3 : The set of all charts φW0,W1in Gk(n) where W0 and W1

runs over all transversal decompositions of Rn with dim(W0) = k is a differen-tiable atlas for the grassmannian Gk(n).

Remark: The formula for the transition function shows that the chartsφW0,W1

forms a real analytic atlas for Gk(n).

Theorem 1.4 : The differentiable atlas in the previous proposition makesGk(n) into a differentiable manifold of dimension k(n− k)Proof: Since every subspace W transversal to W1 is the graph of a linear mapT ∈ Lin(W0,W1) and we have dim(Lin(W0,W1)) = k(n− k).

Examples:

1. G1(n) is the grassmannian of all lines passing through origin in Rn alsocalled RPn−1. For n = 2 we have G1(2) ∼= RP 1 ∼= L(1)

1.2 Tangent space to a grassmannian

In this part we establish the tangent space to the grassmannian at a pointW ∈ Gk(n) is

TW (Gk(n)) ∼= Lin(W,Rn/(W ))

This enables us to compute the derivative of a curve in Gk(n)

Proposition 1.5: Let W ∈ Gk(n) and Rn = W ⊕ W1, denote by q1 :W1 → Rn/(W ) is the restriciton of the quotient map to W1 then we have anisomorhism


Φ = Lin(Id, q1) dφW,W1(W ) : TW (Gk(n))→ Lin(W,Rn/(W )) · · · · · · · · · (1)

where

Lin(Id, q1) : Lin(W,W1)→ Lin(W,Rn/(W )) · · · · · · · · · · · · (2)

is the operator T 7→ q1 TThe isomorphism is canonical in the sense that it does not depend on the choiceof the complementary subspace W1.

Remark: Since the isomorphism is independent of the choice of the com-plementary subspace W1 we can in particular choose it to be the vertical 0×Rn

Proof: We simply observe since φW,W1 : Gk(n,W1) → Lin(W0,W1) is achart the corresponding derivative map dφ is an isomorphism. It remains toshow that the isomorphism is canonical.Observe that q1 is an isomorphism and φW,W1

is a chart around W and hence(1) defines an isomorphismClaim: The isomorphism Φ is independent of the choice of W1

Proof of Claim: Simply choose another complementary subspace W′

1 such thatW ⊕W ′

1 = Rn.

W1

ηWW1,W

′1−→ W

′

1

q1

q′1

Rn/W

We observe that φW,W1(W ) = φW ,W

′1(W ) = 0 the following diagram com-

mutes

TW (Gkn)

dφW,W1(W )

dφW,W

′1

(W )

Lin (W,W1)Lin (Id,ηW

W1,W′1

)

−→ Lin (W,W′

1)

hence we have our desired result.Remark: Henceforth we will identify the spaces TW (Gk(n)) ∼= Lin(W,Rn/(W ))

1.3 The Lagrangian grassmanian

Denote by Λ the set of all lagrangian subspaces of a 2n dimensional sym-plectic space (V, ω)(isomorphic to R2n). Λ is clearly a submanifold of Gn(V )

Definition 1.6: Λ(v, ω) = Λ = L ∈ Gn(V )|Lis a lagrangian is called thelagrangian grassmanian of the symplectic space (V, ω)For W ∈ Gk(V ) we will recall the identification here

TW (Gk(V )) ∼= Lin(W,V/(W ))

Lemma 2.2 Let (L0, L1) be a lagrangian decomposition of V . Then asubspace L ∈ Gn(L1) is lagrangian iff the bilinear form ρL0,L1

φL0,L1(L) ∈

Lin(L0, L∗0) ∼= B(L0) is symmetric.


Proof: Since dim(L) = n, L is a lagrangian iff the symplectic form vanishes onisotropic on L. Let T = φL0,L1(L) so that T ∈ Lin(L0, L1). Since L = Gr(T )we have that

ω(v + Tv,w + Tw) = 0 or ω(v, Tw)− ω(Tv,w) = 0 orω(v, Tw) = ω(Tv,w) orT = T ∗

hence T is a symmetric bilinear map.

Definition 1.8 : If L1 ⊂ V is a lagrangian subspace we denote by Λ(L1)the set of all lagrangian subspaces of V transversal to L1 i.e.,

Λ(L1) = Λ ∩Gn(L1)

the previous lemma has the following important corollary

Corollary 1.9 : Associated to each lagrangian decomposition (L0, L1) of Vwe have a bijection φL0,L1

: Λ(L1)→ Bsym(L0)

Corollary 1.10 : The lagrangian grassmannian is an embedded submani-

fold of Gn(V ) with dimension dim(Λ) = n(n+1)2 . The charts φL0,L1

defined onΛ(L1) form a differntiable atlas for Λ as (L0, L1) runs over all the lagrangiandecompositions of V .Proof: Given a lagrangian decomposition (L0, L1) of V it follows from lemma1.7 that the chart

W 7→ ρL0,L1 φL0,L1(W ) ∈ Lin(L0, L∗0) ∼= B(L0) for all W ∈ Gn(L1)

of Gn(V ) is a submanifold chart for Λ. Moreover dim(Bsym(L0)) = n(n+1)2 .

(Observe that W is a lagrangian iff φ maps onto Bsym(L0))Also domain of the charts φL0,L1

covers all the lagrangians W ∈ Gn(L1) as(L0, L1) runs over all the lagrangian decompositions of V .

We have the following corollary that characterizes the tangent space TL(Λ)of the lagrangian grassmanian.

Corollary 1.11 Let L ∈ Λ be a fixed lagrangian then the isomorphism

Lin(Id, ρl) : Lin(L, V/L)→ Lin(L,L∗) ∼= B(L) given by

Z 7→ ρl Z takes

TL(Λ) ⊂ TL(Gn(V )) ∼= Lin(L, V/L) onto the subspace Bsym(L) ⊂ B(L)

Remark: Using the above isomorphism we will henceforth identify TL(Λ)with the euclidean space Bsym(L). Observe that the dimension of Bsym(L)mathches with the dimension of the tangent space TL(Λ) which is equal to thedimension of Λ.


1.4 The submanifolds Λk(L0)

For a lagrangian L ⊂ V consider the following subsets of ΛΛk(L0) = L ∈ Λ|dim(L ∩ L0) = k, clearly Λ =

⋃nk=0 Λk(L0)

Let us look at the symplectic action on the lagrangian grassmannian.Definition 1.12 Denote Sp(V, ω, L0) as the subset of Sp(V, ω) consisting ofsymplectomorphisms that preserve L0 i.e.,

Sp(V, ω, L0) = A ∈ Sp(V, ω)|A(L0) = L0

Lemma 1.13 : The lie algebra sp(V,w, L0) of the symplectic group Sp(V, ω, L0)consists of those linear endomorphisms X ∈ Lin(V ) such that ω(X., .) is a sym-metric bilinear form that vanishes on L0

Proof: We simply observe that ω(X., .)|L0×L0= 0 iff X(L0) is contained in

the ω orthogonal complement of L0. But L0 is lagrangian hence we have thatL⊥0 = L0

Remark: Clearly the action of Sp(V, ω) leaves each stratum Λk(L0) invari-ant. Also it canbe shown that Λk(L0) is the orbit of an action of Sp(V, ω, L0).Then if we are able to show that Λk(L0) is locally closed in 1 in Λ then by thefollowing theorem it proves that it is an embedded submanifold of Λ

Theorem 1.14: Let G be a Lie group acting differentiably on the manifoldM . Let m ∈ M the orbit G(m) ⊂ M is an embedded submanifold iff G(m) islocally closed in M .

Lemma 1.15: For all k = 0, 1, 2, · · · · · · , n the subset Λ≤k(L0)(⋃ki=0 ΛL0)

and Λ≥k(L0)(⋃ni=k Λi(L0)) are respectively are open and closed subsets of Λ

Obs: Since Λk(L0) =⋂≥k

(L0)⋂

Λ≤k(L0) it follows that Λk(L0) is an em-bedded submanifold of Λ

Proof of lemma 1.15: The set of spaces, W ∈ Gn(V ) such that dim(W⋂L0) ≤

k is open in Gn(V ). Since Λ inherits the subspace topology Λ≤k(L0) is open inΛ and Λ≥k the complement is closed.

Theorem 1.5: For eack k = 0, 1, 2, · · · , n, Λk(L0) is an embedded subman-

ifold of Λ with codimension k(k+1)2 , the tangent space is given by

TL(Λk(L0)) = B ∈ Bsym(L)|B|(L0∩L)×(L0∩L) for all L ∈ Λk(L0)

Proof: Let σ be a fixed lagrangian consider λ ∈ Λk(n) its intersection withσ consider the natural mapping

π : Λk(n)→ Gk(σ)

given by λ 7→ λ ∩ σ. Now without loss of generality assume σ := Rn × 0 thenthe fiber over H is the set.

1S ⊂ X is said to be locally closed in X if S = U ∩ V where U ⊆ X is open and V ⊂ X isclosed

2 A BRIEF REVIEWOF THE THEORYOF SYMMETRIC BILINEAR FORMS7

Λ(n− k) = σ ∈ Λk(n)|dim(λ ∩H) = k

passing onto a frame Z = (X,Y ) for the lagrangian λ we observe the first kcolumns of Z must vanish. Since λ is a lagrangian Z is a symmetric matrixwhich implies that

[Z] =

(0 00 D

)where D is a (n− k)× (n− k) symmetric matrix. Hence

dim(Λ(n− k)) = (n−k)(n−k+1)2

Since dimGk(H) = k(n − k) and Gk(H) = Λk(n)/Λ(n− k) it follows after alittle bit of algebra that

dim(Λk(n)) = dim(Λ(n))− k(k+1)2

proving the first part of the theorem. For the second part of the theorem usinglemma 1.13 we have

TLΛk(L0) = B|L×L|B ∈ Bsym(V ), B|L0×L0= 0 forall L ∈ Λk(L0)

Conversely let B ∈ Bsym(L) such that B|L0∩L = 0 then choose for a fixed Land varying k a family Lα

L0 =⋃Lα

(Lα ∩ L0) where Lα ∈ Λk(L0)

But since B|Lα∩L0= 0∀Lα it proves that B|L0×L0

= 0

2 A brief review of the theory of symmetric bilinear forms

In this short section we want to state some of the important results concerningsymmetric bilinear forms that would be required in the next section of Maslovindex.

Definition 2.1: Let B ∈ Bsym(V ) we say that B is• positive definite if B(v, v) > 0∀v ∈ V, v 6= 0

• positive semi definite if B(v, v) ≥ 0∀v ∈ V

• negative definite if B(v, v) < 0∀v ∈ V, v 6= 0

• negative semi definite if B(v, v) ≤ 0∀v ∈ V

Definition 2.2: The index of B denoted by n−(B) is defined by n−(B) =supdimW |W is B negative subspace of V hence define coindex n+(B) = n−(−B).Define the signature of B by sgn(B) = n+(B)− n−(B).

Remark:

• Recall that B is said to be non degenerate if Ker(B) = v ∈ V |B(v, w) =0∀w ∈ V

• Observe that B is non degenerate if the associated linear operator φ : V →V ∗ given by v 7→ B(v, .) is injective

2 A BRIEF REVIEWOF THE THEORYOF SYMMETRIC BILINEAR FORMS8

We state an important proposition now

Proposition 2.3 If B ∈ Bsym(V ) and V = Z ⊕W with B positive definitein Z and negative definite in W then B is non degenerate.Proof: Take v = v+ + v− from Ker(B) compute B(v, v+) and B(v, v−) andequate both to zero.

The most important result about the theory of symmetric bilinear forms isthe following result due to Sylvester

Theorem 2.4: Sylvester’s inertia theorem: Suppose dimV = n < ∞and let B ∈ Bsym(V ) then there exists a basis of V with respect to the matrixof B is given by

B ∼

Ip 0p×q 0p×r0q×p −Iq 0q×r0r×p 0r×q 0r

where 0α×β , 0α and Iα are the zero and identity matrices of orders α×β, α×α,α× α respectively. The numbers p, q, r are uniquely determined by B

n+(B) = p, n−(B) = q, dgn(B) = r

where dgn(B) = dimKer(B)

We are mainly interested in the evolution of the index of one parameterfamily of symmetric bilinear form.

Let t 7→ B(t) be one parameter family of symmetric bilinear forms on aspace V . Define a norm of B ∈ B(V ) by setting

‖B‖ = sup‖v‖≤1,‖w≤1‖‖B(v, w)‖

Obs: Since B(v) is finite dimensional since V is finite dimensional and anyother norm on B(V ) gives rise to the same topology.

Lemma 2.5 : Let k ≥ 0 be a fixed set of symmetric bilinear forms B ∈Bsym(V ) such that n−(B) ≤ k is open in Bsym(V )Proof: Let B ∈ Bsym(V ) with n−(B) ≥ k then since

n−(B) = supdimW |WisB negative

then ∃ a k dimensional B negative subspace W ⊂ V . But unit sphere Ω ⊂ Wis compact we have sup‖v‖=1,v∈WB(v, v) = c < 0 then if we choose A such that‖A−B‖ < c/2 we have that n−(A) ≥ k which proves that the set is open.

Corollary 2.6 : Let k ≥ 0 be fixed then the set of non degenrate symmetricbilinear forms such that n−(B) = k is open on Bsym(V )

Corollary 2.7: Let t 7→ B(t) be a continuous curve defined in Bsym(V ) insome interval I ⊂ R. If B(t) is non degenerate for all t ∈ I then n−(B(t)) andn+(B(t) are constant in I

3 MASLOV INDEX FOR LAGRANGIAN PATHS 9

Remark: The above corollary shows that the only way the signature of acontinuous one parameter family of symmetric bilinear forms can change is ifthere exists a t0 ∈ I such that B(t0) is degenerate.

Theorem 2.8: Let B : [t0, t1] → Bsym(V ) be a curve of class C1. Let

N = KerB(t0). Supose that the bilinear form B′(t0)|N×N is non degenerate

then there exists ε > 0 such that for t ∈ (t0, t0 + ε) the bilinear form B(t) isnondegenerate and

n+B(t) = n+B(t0) + n+B′(t0)|N×N

n−B(t) = n−B(t0) + n−B′(t0)|N×N

We will not prove this theorem but use it in our next section on Maslov index.

3 Maslov index for lagrangian paths

We employ a slight change of notation here for convenience. We denote byL(n) the lagrangian grassmannians in R2n. We have already seen that for a fixedlagrangian subspace V ⊂ R2n we have the decomposition L(n) =

⋃nk=0 Σk(V ).

The maslov cycle determined by V is the algebraic varietyΣ(V ) = Σ1(V ) =

⋃k≥1 Σk(V )

We recall theorem 1.16 here to restate the isomorphism

TΛΣK(V ) = Q ∈ Bsym(V )|Q|Λ∩V = 0

between the tangent space at Λ and the space of all symmetric bilinear formsthat vanish on Λ∩V . We state an important theorem that allows us to calculatethe symmetric bilinear form Q in general

Theorem 3.1: Let Λ(t) ∈ L(n) be a curve of lagrangian subspaces withΛ(0) = Λ and Λ(0) = Λ then

1. Let W be a fixed lagrangian complement of Λ and for small t define w(t) ∈W by v + w(t) ∈ Λ(t) then the form defined by

Q(v) = ddt |t=0

ω(v, w(t))

is independent of the choice of W

2. The form Q is natural in the sense that

Q(ΨΛ,ΨΛ) = Q(Λ, Λ) for a symplectic matrix Ψ ∈ Sp(2n)

Proof: Choose Λ(by rotating the coordinate frame if required) such that Λ(0) =Rn × 0 and for small t, Λ(t) = (x,A(t)x)|x ∈ Rn where A(t) ∈ R(n × n) issymmetric. Hence we have v = (x, 0), w(t) = π1(Λ(t)) = (0, A(t)x) where π1 isthe usual projection along the vertical then we haveω(v, w(t)) = < (x, 0), (0, A(t)x) > = < (x,A(t)x) >. So

Q(v) = ddt < (x,A(t)x) > = < (x, A(0)x) >


So Q is independent of the choice of the complement W .

Let Ψ =

(A BC D

)be a symplectic matric of dimension 2n × 2n. We claim

that

Q(ΨΛ,ΨΛ) Ψ)v = Q(Λ, Λ)v

. Let V = Ψ(v) = (Av + bL(t)v, Cv + DL(t)v) where we have taken Λ(t) =Gr(L(t)). Now its aa straightforward computation to calculate

Q(ΨΛ,ΨΛ) Ψ)v =d

dt |t=0ω((Av + bL(t)v, Cv +DL(t)v), (0, Cv +DL(t)v))

=d

dt |t=0(Av + bL(t)v, Cv +DL(t)v)

= < v,ATDL(0)v > + < CTBL(0)v, v > (Observing thatL(0)v = 0)

= < v, L(0)v > (since Ψ is symplectic we haveATD − CTB = 1)

= Q(Λ, Λ(0))

Definition 3.2: Crossing Let Λ : [a, b] → L(n) be a smooth curve oflagrangian submanifolds. A croosing for Λ is a number t ∈ [a, b] such thatΛ(t) ∩ V 6= φ define the crossing form at the crossing by

Γ(Λ, V, t) = Q|Λ(t)∩V

A crossing is regular if Q is non singular at the point it is simple if Λ(t) ∈ Σ1(V ).From theorem 3.1 part 2 we have that

Corollary 3.3 : Crossing form at a crossing t ∈ [a, b] is natural i.e.,

Γ(ΨΛ,ΨV, t) Ψ = Γ(Λ, V, t)

for every symplectic matrix Ψ ∈ Sp(2n)

Remark: It is immediately observed that if we take a lagrangian path ofsymmetric matrices Λ(t) = Gr(A(t)) then at each crossing the crossing formhas the expression

Γ(Λ, V, t) = < x, A(t)x >

with x ∈ Ker(A(t))

Definition 3.4 : Maslov index For a curve Λ : [a, b] → L(n) with onlyregular crossings define the Maslov index for the path Λ(t) relative to V by

µ(Λ, V ) = 12sgnΓ(Λ, V, a) + Σa<t<bsgnΓ(Λ, V, t) + 1

2sgnΓ(Λ, V, b)

where t runs over all crossings

The most important properties of Maslov index are given by the followingaxioms


Theorem 3.5:

• Naturality For Ψ ∈ Sp(2n) µ(ΨΛ,ΨV ) = µ(Λ, V )

• Catenation for a < c < b µΛ, V = µ(Λ|[a,c], V ) + µ(Λ|[c,b], V )

• Localization If Λ(t) = Gr(A(t)) then µ(Λ, V ) = 12sgnA(b)− 1

2sgnA(a)

• Homotopy two paths Λ0 and Λ1 with Λ0(a) = Λ1(a) and Λ0(b) = Λ1(b)are homotopic iff they have the same Maslov index relative to a fixedlagrangian V

• Zero: Every path Λ : [a, b]→ Σk(V ) has µ(Λ, V ) = 0 for every lagrangianV

Obs: We have already proved the naturality axiom. Now we prove the lo-calization and the homotopy axiom and postpone the proof of the zero axiomfor the next subsection

Proof of Localization axiom: It can be shown that the regular crossingsare isolated and form a compact subset of the real interval [0, 1] hence it is afinite set. Let t0, t1, · · · · · · , tn be the finite set of crossings then we have bytheorem 3.8

sgnA(t0 + ε) = sgnA(t0) + sgnΓ(Λ, V, t0) · · · · · · · · · · · · (1)

Now by the Maslov index formula and by the above formula (1) taking ε =(t1 − t0) we get that

µ(Λ, V ) =1

2sgnΓ(Λ, V, t0) +

1

2Σn−1i=1 sgnΓ(Λ, V, ti) +

1

2sgnΓ(Λ, V, tn)

= 1

2sgn(A(t1)− sgnA(t0))+

1

2Σn−1i=0 sgnΓ(Λ, V, ti) +

1

2sgnΓ(Λ, V, tn)

= −1

2sgnA(t0) +

1

2sgnA(t2) +

1

2Σn−1i=2 sgnΓ(Λ, V, ti)

1

2sgnΓ(Λ, V, tn)

( Since sgnA(t1) + sgnΓ(Λ, V, t1) = sgnA(t2))

= −1

2sgnA(t0) +

1

2(A(tn))(proceeding like this after n− 1 steps)

Proof of the homotopy axiom: Without loss of generality we can assumethat we can approximate a path of lagrangian with one having only regularcrossings at the endpoint and simple crossings at the interior. Then considerΛ0 ∼H Λ1 where H is a path homotopy.Also by the naturality axiom assume that the endpoints are transversal henceare graphs of symmetric matrices. Now by the homotopy choose ε > 0 smallsuch that Λj(t) ∈ Σ0(V ) and Λs(t) is a graph for a < t ≥ a+ ε and b− ε ≤ t < bsuch that

s 7→ Λs(a+ ε) and s 7→ Λs(b− ε)

the paths have simple crossings at the interior. By the localization axiom wehave the Maslov index for these paths as

µ(s 7→ Λs(a+ ε)) =1

2sgnΓ(Λ1, V, a)− 1

2sgnΛΓ(Λ0, V, a)

µ(s 7→ Λs(b− ε)) =1

2sgnΓ(Λ0, V, b)−

1

2sgnΓ(Λ1, V, b)


Introduce the intermediate path Λ′

which is the catenation of the pathswhich is the catenation of the paths Λ0|[a, a+ ε],Λs(a+ε),Λ1|[a+ ε, b− ε],Λ1−s(b−ε),Λ0|[b− ε, b]. Since all three paths Λ0,Λ

′,Λ1 have the same Maslov index.

Corollary 3.6: For Λ(t) = Λ(t+ 1) a loop of lagrangian subspaces. And forany lagrangian V

µ(Λ, V ) = α(1)−α(0)2π

where det(X(t)+ iY (t)) = eια(t) then Z(t) = (X(t), Y (t)) being a unitary framefor Λ(t)Proof: We simply observe the case n = 1. Since Λ(1) = RP 1 by naturalityif we choose V = L0 crossing can occur only at t = 0, 1/2, 1 and choose thefunction α such that α is monotone increasing without self crossing

Fig 2: Rotating lagrangian frames in dimension n = 1

α : [0.2π]→ Rt 7→ 2πt

then

µ(Λ, V ) =1

2[sgnΓ(Λ, V, 1) + sgnΓ(Λ, V, 0)] + sgnΓ(Λ, V, 1/2)

=1

2[sgn < X(1)u, Y (1)(u) > +sgn < X(0)u, Y (0)(u) >] + sgn < X(1/2)u, Y (1/2)u >

=1

2(sgnα(0) + sgnα(1)) + sgnα(1/2)

=1

2(1 + 1) + 1

= 2

=α(1)− α(0)

2π

Remark: We see that Maslov index for the loop equals the rotation number(winding number) for the loop for the case n = 1. For higher dimensional casethis equivalence is expressed in terms of the homological index of the loop.

3.1 Maslov Index for lagrangian pairs

Definition 3.7: Consider a pair of curves Λ,Λ′

: [a, b] → L(n) define the rela-tive crossing form on Λ(t) ∩ Λ

′(t)


Γ(Λ,Λ′, t) = Γ(Λ,Λ

′(t), t)− Γ(Λ

′,Λ(t), t)

for a pair with only regular crossings define the relative Maslov index by

µ(Λ,Λ′) = 1

2sgnΓ(Λ,Λ′, a) + Σa<t<bsgnΓ(Λ,Λ

′, t) + 1

2sgnΓ(Λ,Λ′, b)

Obs: In case Λ′(t) = V for all t i.e, constant lagrangian by naturality choose

V = Rn × 0 then the second term in the relative crossing form is

Γ(Λ′,Λ(t), t) = Γ(V,Λ(t), t)

=d

dt |t=0ω((x, 0), (0, 0))

= 0 (π1(x, 0) = (0, 0))

hence agrees with µ(Λ, V )

Theorem 3.8 : The relative maslov index is natural i.e., µ(ΨΛ,ΨΛ′) =

µ(Λ,Λ′)

Proof: We have already shown the crossing form is natural i.e.,

Γ(ΨΛ,ΨΛ′, t) Ψ(t) = Γ(Λ,Λ

′, t)

Now we observe that Q(ΨΛ,ΨΛ′, t)|Ψ(Λ(t))∩Λ′ (t) = Q(ΨΛ

′,ΨΛ, t)|Ψ(Λ(t))∩Λ′ (t) so

they agree on the intersection hence they always agree.

Theorem 3.9: Consider the symplectic vector space (R2n×R2n, (−ω)×ω),then

µ(ΨΛ,Λ′) = µ(Gr(Ψ),Λ× Λ

′)

in particular when Ψ(t) = 1 we have

µ(Λ,Λ′) = µ(4,Λ× Λ

′) and 4 ⊂ R2n × R2n is the diagonal

Proof: We want to prove the following claimClaim: Γ(4,Λ× Λ

′) = Γ(Λ,Λ

′(t), v)− Γ(Λ

′,Λ(t), v)

Proof of Claim: We work in the product space with Λ(t) = Λ(t)∩Λ′(t), v = (v, v)

with v ∈ Λ(t) ∩ Λ′(t) and the product form is given by ω = (−ω)× ω

Choose a lagrangian complement W = W ×W ′such that both W ∩Λ(t) = φ

and W′ ∩ Λ

′(t) = φ then we have

v + w(s) ∈ Λ(s), v + w′(s) ∈ Λ

′(s) where w(s) ∈W and w

′(s) ∈W ′

then define w(s) = (w(s), w′(s)) ∈W . Aplying the symlectic form on w we get

that

ω((v), w(s)) = −ω(v, w(s) + ω(v, w′(s))

differentiating the identity with respect to s at s = t we have


Γ(4,Λ× Λ′, t)v = Γ(Λ,Λ

′(t), t)v − Γ(Λ

′,Λ(t), t)v

For proving the first identity we define the sympelctomorphism

Ψ : (R2n × R2n, ω)→ (R2n × R2n, ω)

defined by (z, z′) 7→ (z,Ψ(t)z

′) then we have

Gr(Ψ) = Ψ(4)

and Ψ(Λ×Ψ−1(Λ′)) = Λ× Λ

′hence we have

µ(Gr(Ψ),Λ× Λ′) = µ(Ψ(4),Ψ(Λ×Ψ−1Λ

′))

= µ(4,Λ×Ψ−1Λ′))(by Naturality)

= µ(Λ,Ψ−1(Λ′))

= µ(ΨΛ,ΨΨ−1Λ′)(by Naturality)

= µ(ΨΛ,Λ′)

Next we consider homotopies of lagrangian pairsCorollary 3.10: The number µ(Λ,Λ

′) is a homotopy invariant in the case that

the end points are transversal to each other i.e.,

Λ(a) ∩ Λ′(a) = φ and Λ(b) ∩ Λ

′(b) = φ

Conversely if two lagrangian pairs are homotopic in this sense they have thesame Maslov index

Proof: Let H and G be the required homotopies that preserve the transver-sality condition on the endpoints.

Fig 3: Homotopy of lagrangian pairsBy theorem 3.9 we have that

µ(Λi,Λ′

i) = µ(4,Λi × Λ′

i)

µ(Λf ,Λ′

f ) = µ(4,Λf × Λ′

f )

Observe that Λi×Λ′

i ∼H×G Λf ×Λ′

f where H×G is the product homotopy.

given by H0(t)×G0(t) = Λi(t)×Λ′

i(t) and H1(t)×G1(t) = Λf (t)×Λ′

f (t) hencewe have


µ(4,Λi × Λ′

i) = −µ(Λi × Λ′

i,4)

= −µ(Λf × Λ′

f ,4) (by homotopy axiom)

= µ(4,Λf × Λ′

f )

⇒ µ(Λi,Λ′

i) = µ(Λf ,Λ′

f )

Conversely let µ(Λ0,Λ′

0) = µ(Λ1,Λ′

1) such that the transversality conditionon the endpoints holds

Fig 4: Stable stratum homotopy between lagrangian pairsUsing the transitive action of Sp(2n) on L(n) choose a fixed lagrangian V

such that

Ψ0(t)Λ′

0(t) = V and Ψ1(t)Λ′

1(t) = V

Then let Ψ0(a)Λ0(a) = W and Ψ1(b)Λ1(b) = V . By the transversality on theendpoints we have that W∩V = φ and W

′∩V = φ. So as µ(Λ0,Λ′

0) = µ(Λ1,Λ′

1)we have that µ(Ψ0Λ0, V ) = µ(Ψ1Λ1, V )Now consider the paths Ψ0Λ0 and Ψ1Λ1, they have the same Maslov index withrespect to the lagrangian V they are both transverse to V so

Ψ0Λ0 ∼Ω Ψ1Λ1

where Ω is a stratum homotopy(see next section) hence the required homotopypairs are (Ωs,Ω

′

s) where Ω′

s(t)Ψs(t)−1V

Now we prove the zero axiom for Maslov indexCorollary 3.11: If Λ : [a, b]→ Σk(V ) then µ(Λ, V ) = 0Proof: Using the naturality axiom and the symplectic action of Sp(2n) on L(n)we choose a path Ψ : [a, b]→ Sp(2n) such that

Ψ(t)Λ(t) = L and Ψ(t)V = L′

where both L,L′

are constant lagrangians

hence we have µ(Λ, V ) = µ(Ψ(t)Λ,Ψ(V )) = µ(L,L′) = 0

Obs:The crucial fact employed in the proof is that the symplectic action onL(n) fixes the kth stratum Σk(V ) for each k.

3.2 Maslov Index for Symplectic paths

Definition 3.12: For a path of symplectic matrices Ψ : [a, b] → L(n) and thefixed lagrangian V to be the vertical 0 × Rn define the maslov index of thesymplectic path Ψ


µ(Ψ) = µ(ΨV, V )

Define the k stratum by

Spk(2n) = Ψ ∈ Sp(2n)|dim(ΨV ∩ V ) = k

Then consider the principal bundle

St(2n)→ Sp(2n)π→ L(n) where π(Ψ) = ΨV

it is immediately clear that St(2n) = Ψ ∈ Sp(2n)|B = 0 and that π−1(Σk(V )) =Spk(2n)

Obs: The subspace Vk = ΨV ∩V of the vertical 0×Rn being k dimensionali.e.,

Vk = v ∈ Rn|v1 = v2 = · · · = vk = 0

is equivalent to the condition that rank(B) = n− k

Lemma 3.13: If Ψ =

(A BC D

)is a symplectic matric then (B,D) is a frame

for the lagrangian Gr(Ψ) with the crossing form

Γ(Ψ, t) : Ker(B(t))→ R

given by

Γ(Ψ, t)(y) = − < D(t)y, B(t)y >

Proof: Let Z(t) = (B(t), D(t)) be a frame for the symplectic path Ψ(t) ∈Sp(2n) then Ψ(t)V = (B(t)v,D(t)v)|v ∈ 0 × Rn taking orthogonal comple-ment along the horizontal we get w(t) = (B(t)v, 0), then

Γ(Ψ, t)(v) = QΨ(t)V ∩V

=d

dt |t=0ω((B(t)v,D(t)v), (B(t)v, 0))

= − d

dt |t=0< D(t)v,B(t)v >

= − < D(0)v, B(0)v > (sinceB(0)v = 0)

at any time t therefore the crossing form takes the formula

Γ(Ψ, t)v = − < D(t)v, B(t)v >

Our next aim is to prove the connectedness of the k stratum Σk(V ), towardsthis we defineDefinition 3.14: Let Fk(n) = Z = (X,Y )|Z is a lagrangian frame and rank(X) =n− k. Define the continuous mapping

ν : Fk(n)→ Z2

defienes the parity for the frame Z = (X,Y ) by


(−1)ν(Z) = sgn det(X1)sgn det(Y4)

where X =

(X1 00 0

)and Y =

(Y1 0Y3 Y4

)are normal form decompositions of

ZWe partition the set of lagrangian frames by an equivalence relation ∼. CallZ ∼ Z1 if there exists positive definite matrices L,M ∈ Rn×n such that

X1 = LTXM and Y1 = L−1YM

Obs:

1. ∼ defines an equivalence relation on Fk(n)

2. Every Lagrangian frame Z is equivalent to one in normal form, hence wecan work explicitly with frames in normal form

3. if Z ∼ Z1 are equicvalent normal forms then (−1)ν(Z) = (−1)ν(Z1)

Claim 3.15: The set of lagrangian frames with +ve and −ve parity areconnected subsets of Fk(n)Proof: Because a lagrangian frame Z = (X,Y ) with +ve parity can be con-nected to the frame with normal form X1 = 1, Y = 1. Similarly the framewith −ve parity can be connected with the frame with normal form X1 =diag(1, 1, · · · ,−1) and Y = 1. Clearly this proves that Fk(n) has two con-nected components with +ve and −ve parity respectively.

Corollary 3.16: Each stratum Σk(V ) is connected.

Proof: Define the map Σk(V )π→ Fk(n)

ν→ Z2 where pi is the map that makesthe following diagram commutative

Σk(V )π−→ Fk(n)

↑ λ = ΨVπ

Spk(2n)

i.e., π λ = π. Since both π and λ are continuous π is continuous. Now thecontinuuity of the function

ν = ν π : Σk(V )→ Z2

claims that every lagrangian Λ ∈ Σk(V ) must have constant parity but this isa contradiction since we can always choose frames ZΛ and Z

′

Λ for Λ such that

(−1)ν(ZΛ) = 1, (−1)ν(Z′Λ) = −1

Obs: For the dimension n = 1 since L(n) ∼= RP 1 is the set of all linesthrough the origin choosing frames of distinct parity amounts to rotating theline by π or Z

′

Λ = eiπZΛ

Definition 3.17 : A homotopy Ψs : [a, b] → Sp(2n) of symplectic paths iscalled a stratum homotopy if the ranks of the endpoints Bs(a) and Bs(b) doesnot change with s.


Obs: for a path of symplectic shear Ψ(t) =1 B(t)0 1

. Since Ψ(t) is symplectic

we observe that B(t) is symmetric B(t)T = B(t). Then ΨV = (B(t)v, v) =Gr(B(t)). So by the localization axiom we have the Maslov index of Ψ

µ(Ψ) = µ(Ψ(t)V, V )

= µ(Gr(Ψ), V )

=1

2(sgnB(b)− sgnB(a))

Remark: Call such a symplectic shear path neutral if

• µ(Ψ) = 0(this will happen if the family B(t) is non degenrate for all t)

• B(a), B(b) ∈ Sp0(2n) and n ∈ 4Z

Definition 3.18 : Two paths Ψ0 and Ψ1 are stably stratum homotopic ifthere exists neutral paths Ψ

′

0 and Ψ′

1 such that Ψ0 ⊕Ψ′

0 ∼H Ψ1 ⊕Ψ′

1 where His a stratum homotopy.

Remark:

• It is straightforward to see that stable stratum homotopy is an equivalencerelation

• For any symplectic path Ψ : [a, b] → Sp(2n) define ka = ka(Ψ) andkb = kb(Ψ) by Ψ(a) ∈ Spka(2n), Ψ(b) ∈ Spkb(2n), νa = νa(Ψ) = ν(Ψ(a)).Clearly under stable stratum homotopy µ, ka, kb, νa, νb remain invariant.

We state the defining theorems of this section now.

Theorem 3.19: Two paths in Sp(2n) are stratum homotopic iff they havethe same invariants µ, ka, kb, νa, νb, these invariants are related by

µ+ ka−kb2 ∈ Z · · · · · · · · · · · · (A)

and

µ+ ka−kb2 ≡ νa − νb(mod2) · · · · · · · · · · · · (B)

Theorem 3.20: Two symplectic paths Ψ0 : [a, b] → Sp(2n0) and Ψ1 :[a, b]→ Sp(2n1) are stably stratum homotopic if and only if n1 − n0 ∈ 4Z andthey have the same invariants µ, ka, kb, νa, νb

Theorem 3.21: Every symplectic path is stably stratum homotopic to a sym-plectic shear.

Proof: We divide the proof in six steps.

Step 1: A symplectic shear satisfies equations A and BAssume Ψ(t) is a symlectic shear with right upper block B(t) , since Ψ(b) ∈Spkb(2n) we have dim(Ψ(b)V ∩ V ) = kb implying that the rank of B(b) =n − kb ⇒ n+

b + n−b = n − kb ⇒ n+b = n − n−b − kb , the number of positive


eigenvalues. Similarly for B(a). Now since νa ≡ n−a (mod2) and νb ≡ n−b (mod2)hence

µ =1

2(sgn B(b)− sgn B(a))

=1

2(n+

a − n−a )− (n+b − n

−b )

=1

2(n− 2n−a − ka)− (n− 2n−b − kb)

=1

2(kb − ka) + (n−b − n

−a )

Step 2: If n, ka, kb ∈ Z, νa, νb ∈ Z2 and µ ∈ R and 0 ≤ ka, kb ≤ n and

|µ| < n − 1 − ka+kb2 then there exists a symplectic shear in Sp(2n) with these

given invariantsTake B(a) and B(b) to be diagonal matrices with entries 0, 1,−1 such that B(a)has ka zeroes and n−a signs with n−a ≡ νa(mod2), similarly for B(b) by step 1,Ψ is precisely the symplectic shear with the given invariants

Step 3: Two paths in Sp(2n) are stratum homotopic iff they have the sameset of invariants, (µ, ka, kb, νa, νb)Since they have the same set (µ, ka, kb, νa, νb) and Σk(V ) is connected the stra-tum homotopy reduces to a path homotopy wih fixed endpoints. Since theyhave the same Maslov index µ by homotopy axiom they are homotopic.

Step 4: Let Ψ0 : [a, b] → Sp(2n0) and Ψ1 : [a, b] → Sp(2n1) be stably stra-tum homotopic then all the invariants are obviously the same hence n1 = n0

etc.Conversely assume that Ψ0 : [a, b] → Sp(2n0) and Ψ1 : [a, b] → Sp(2n1) havethe same invariants and n1 − n0 ∈ 4Z Now choose neutral paths Ψ

′

0 and Ψ′

1

then the Maslov index

µ(Ψ0 ⊕Ψ′

0) = µ(Ψ0) + µ(Ψ′

0)

= µ(Ψ0) (since Ψ′

0 is neutral)

= µ(Ψ1)

= µ(Ψ1 ⊕Ψ′

1)(since Ψ′

1 is neutral)

Clearly the rest of the conditions for stable stratum homotopy are preserved.

Step 5: Let Ψ : [a, b]→ Sp(2n0) be a symplectic path with maslov index µ.

Choose Ψ′ ∈ Sp(2n′) neutral such that n

′> 2µ (i.e., µ is bounded above by

n′), then by Step 2 we can find a symplectic path with the same invariants as

Ψ

Step 6: Let Ψ : [a, b] → Sp(2n0) be a symplectic path , then we know that

Ψ ∼ Ψ′

where Ψ′

is a symplectic shear, so Ψ has the same invarianrts as Ψ′.

4 ARNOLD’S INTERPRETATION AND THE BOUNDARYMASLOV INDEX20

But Ψ′

satisfies equation A and B of theorem 3.19 hence Ψ also satisfiesWe sum up the discussion with the following important corollary

Corollary 3.21: Two lagragian paths are stratum homotopic with respectto V iff µ, ka, kb are same for both and these invariants are related by

µ+ ka−kb2 ∈ Z

Proof: Since µ(ΨV, V ) = µ(Λ, V ) and µ(Ψ′V, V ) = µ(Λ

′, V ) for some paths

Ψ,Ψ′ ∈ Sp(2n) define the lift

L(n)→ Sp(2n)

by

Λ 7→ Ψ Λ′ 7→ Ψ

′

. Now for the paths Ψ and Ψ′

the invariants µ(since it is invariant underhomotopy) and ka, kb (because for a frame Z = (B,D) rank of B will notchange under the lift) does not change. So it implies that

(µ+ ka−kb2 )Ψ ∈ Z

but since Ψ ∼ Ψ′

so we must have

(µ+ ka−kb2 )Ψ′ ∈ Z

So their projections ΨV and Ψ′V also satisfy the equations above. The converse

arguement is identical.

4 Arnold’s interpretation and the boundary Maslov index

In this brief section we present Arnold’s classical definition of maslov index asthe degree of a map. Consider

R(n) := Gl(n,C)/Gl(n,R)

be the manifold of totally real subspaces of Cn. The lagrangian grassmannianis a submanifold of R(n)

L(n) := U(n)/O(n) = Sp(2n)/Gl(n,R) ⊂ R(n)

Define the map ρ : R(n)→ S1 by

ρ(a.Gl(n,R)) := deta2

det(a∗a)

Let Γ be the compact oriented manifold without boundary i.e., the disjointunion of circles. The Maslov index of a map Λ : Γ→ R(n) is defined by

µ(Λ) := deg(ρ Λ)

Theorem 4.1 Arnold: Two loops in R(n) are homotopic iff they have thesame Maslov index

Fact: Any complex line bundle E over S1 is trivialProof: Choose any trivialization of the bundle such that the transition func-tion gαβ : Uα ∩ Uβ → GL(1,C) ∼= C∗ but the map s : S1 → E given bys(x) = (x, gαβ(x)v) where v ∈ Ex the fiber over x gives a nowhere zero sectionof the bundle In fact its not difficult to show that any complex vector bundleover S1 is trivial.


4.1 Boundary Maslov index

Definition 4.2 A bundle pair (E,F ) over Σ(a compact Riemann surface pos-sibly with boundary) consists of a complex vector bundle E → Σ and a totallyreal subbundle F ⊂ E|∂Σ over the boundary.

A decomposition of a bundle pair (E02, F02) consists of two bundle pairs(E01, F0 ∪ F1) over Σ01 and (E12, F1 ∪ F2) over Σ12 such that

Σ02 = Σ01 ∪ Σ12

and

Fi ⊂ E02|∂Σ01∩∂Σ12

For notational simplicity denote the intersection of the boundary of the decom-posing manifolds as Γ = ∂Σ01 ∩ ∂Σ12. Γ is a compact oriented closed manifoldhence is S1 or a disjoint union of copies of S1.

The next theorem defines the boundary Maslov index axiomatically.

Theorem 4.3: There is a unique operation called the boundary Maslov in-dex that assigns an integer to each bundle pair (E,F ) and satisfies the followingaxioms

• Isomorphism: If Φ : E1 → E2 is a vector bundle isomorphism covering adiffeomorphism φ : Σ1 → Σ2 then we have that

µ(E1, F1) = µ(E2,Φ(F1))

• Direct Sum:µ(E1 ⊕ E2, F1 ⊕ F2) = µ(E1, F1) + µ(E2, F2)

• Composition: For a bundle pair decomposition as in the previous definitionwe have

µ(E02, F02) = µ(E01, F01) + µ(E12, F12)

• Normalization: For Σ = D the unit disc, and E = D×C the trivial bundleand Fz = Reikθ/2, z = eiθ ∈ ∂D = S1 we have

µ(D × C, F ) = k

• Trivial bundle: if ∂Σ = φ and E = Σ× Cn we have

µ(E,F ) = µ(Λ)

where Λ : ∂Σ→ R(n) is defined by Λ(z) := Fz

• Chern Class: If ∂Σ = φ then

µ(E, φ) = 2〈c1(E), [Σ]〉 = 2∫

Σc1(E)

where c1(E) ∈ H2(Σ) is the first chern class evaluated in the fundamentalclass of the compact Riemann surface Σ


The theorem that characterizes the isomorphism axiom is

Theorem 4.4: Two bundle pairs (Ei, Fi) i = 1, 2 are isomorphic over thesame riemann surface Σ iff the following conditions hold

• E1 and E2 have the same rank

• µ(E1, F1) = µ(E2, F2)

• The subbundles F1|C = F2|C for each component C of the boundaryΓ =

⋃C

Remark: The trick in proving the axioms and the theorem is to isolate thetotally real subbundle F from the total bundle E. To this we recall the partialframing of a bundle of real rank n.Let F → Γ be a real rank n bundle then F splits as the Whitney sum F = L⊕F ′

where L is real line bundle and a rank n − 1 bundle F′. The following lemma

simply formalizes what we want to do with a framingA framing is just a choice of basic sections s1, s2, · · · , sn−1 such that they forma basis at each point of Γ

Lemma 4.5: Let ∂Σ 6= φ and let (E,F ) be a bundle pair over Σ with apartial framing (L,F

′, s1, s2, · · · sn−1) then there is a trivialization Φ : E →

Σ× Cn and a map λ : ∂Σ→ S1 such that

Φ(Lz) =√

(λ(z))Re1, Φ(si)(z) = ei for i = 1, · · · , n− 1

where ei is the standard basis of Cn. In particular Φ(Fz) =√λ(z)R×Rn−1, z ∈

∂ΣWe will not prove the lemma but use in the following immediate corollary

of it

Corollary 4.6 : For any rank one bundle pair (E,F ) over the disc D thereis a trivialization Φ : E → D×C such that Φ(Fz) = Rekθ/2 for all z ∈ ∂D = S1

Proof: Simply define the function λ(z) = eikθ in the above corollary.

Outline of the proofs of theorem 4.3 and 4.4: To show existence ofthe boundary Maslov index operator we define it as given by the axioms oftheorem 4.3 and show that it is independent of the choice of the trivializationΦ as required by the definition Recall that if ∂Σ 6= Φ and (E,F ) is a bundlepair over Σ we have defined

µ(E,F ) = µ(ΛΦ)

where ΛΦ(z) = Fz where Fz being an element of the lagrangian grassmannianR(n).Claim: The definition of the boundary Maslov index is independent of thechoice of the trivialization ΦProof: Let Ψ : E → Σ × Cn be another trivialization determined by thetransition function U : Σ→ Gl(n,C). such that

Φ Ψ−1(z, v) = (z, U(z)v), z ∈ Σ, v ∈ Cn

and


U(z)ΛΨ(z) = ΛΦ(z)

Let a ∈ Gl(n,C) satisfy ΛΨ(z) = a(Rn) then U(z)aRn = ΛΦ(z) hence by thedefinition of the map ρ we have

ρ(Λφ(z)) = g(z)ρ(ΛΨ(z)) where g(z) = det(U(z)2)detU(z)∗U(z)

for z ∈ ∂Σ but g : ∂Σ→ S1 as a map extends to Σ hence has degree zero(sinceD is contractible) so µ(ΛΦ) = deg(ρ ΛΦ) = deg(ρ ΛΨ) = µ(ΛΨ)

Remark on the Composition axiom: When the manifold Σ is not closedi.e., has a boundary we give ∂Σ a orientation that it iherits from Σ, now for thedeccomposing surfaces Σ01 and Σ12 the condition ∂Σ01∩∂Σ12 = ∂Σ means thaton the intersection µ(Λ01) = −µ(Λ12) hence they cancel out, for the indepen-dent boundary components we have the algebraic sum µ(Λ02) = µ(Λ01)+µ(Λ12)

Theorem 4.7: Suppose ∂Σ = φ Let Σ = Σ0 ∪ Σ1 be a decomposition ofΣ define Γ = ∂Σ0 = ∂Σ1 = Σ0 ∩ Σ1. Suppose E is a vector bundle over Σ,F ⊂ E|Γ is a totally real subbundle and let Ei = E|Γi then

2〈c1(E), [Σ]〉 = µ(E0, F ) + µ(E1, F )

Proof: Using the lemma 4.5 we conclude that E splits as a line bundle and atrivial bundle of rank n − 1, Σ × Cn−1. Then the real subbundle intersects asF ∩ (Γ× Cn−1) = Γ× Rn−1.Assume without loss of generality n = 1. A unitary trivialization Φ : E|Σi →Σi × C has the form Φ−1

i (z, v) = vsi(z) for z ∈ Σi and v ∈ C and si is a basicsection of Σi. Define u : Γ→ S1 by

s1(z) := u(z)s0(z), z ∈ Γ

Also define the lagrangian subspaces Λ0(z) := u(z)Λ1(z) for z ∈ Γ hence

ρ(Λ0) = u2ρ(Λ1)

Now orient Γ as the boundary of Σ0 hence the Maslov index µ(E0, F ) = deg(ρΛ0) and by the remark on the composition axiom µ(E1, F ) = −deg(ρΛ1) henceby taking degrees we get

µ(E0, F ) + µ(E1, F ) = 2deg u

Let∇ be a hermitian connection for E and define α0 := s−10 ∇s0 ∈ Ω1(Σ0, iR)

and α1 := s−11 ∇s1 ∈ Ω1(Σ1, iR)

5 CAUCHY RIEMANNOPERATORS ANDAPPLICATIONOF THE RIEMANN-ROCH FORMULA24

Then F∇|Σ0= dα0 and F∇|Σ1

= dα1, (α1 − α0)|Γ = u−1du hence

〈c1(E), [Σ]〉 =i

2π

∫Σ

F∇

=i

2π(

∫Σ

α0 +

∫Σ

α1)

=i

2π

∫Γ

(α0 − α1)

=1

2πi

∫Γ

u−1du

= deg u

5 Cauchy Riemann operators and application of the Riemann-Rochformula

We will briefly take a detour to Cauchy Riemann operators first and recall someFredholm regularities of such operators. Then we proceed to state the Riemann-Roch boundary value problem in terms of real linear Cauchy-Riemann operatorson vector bundles over a compact Riemann surface Σ. In the last subsection wewill briefly discuss Hofer Lizan and Sikorav’s paper to calculate the dimensionof the space of pseudoholomorphic curves in an almost complex manifold ofdimension 2n ≥ 4

5.1 Introduction to Cauchy-Riemann operators

Let Σ be a compact Riemann surface with boundary and E → Σ a smoothcomplex vector bundle. Denote by

• Ωk(Σ): smooth complex k forms on Σ(sections of the bundle ΛkT ∗Σ⊗C)

• Ωp,q(Σ): those of type (p, q) (sections of the bundle Λp,qT ∗Σ⊗ C)

• Ωk(Σ, E): the smooth E valued k forms on Σ(sections of the bundleΛkT ∗Σ⊗C E)

• Ωp,q(Σ, E)(sections of the bundle Λp,qT ∗Σ⊗C E)

A complex linear smooth Cauchy-Riemann operator on the bundle E → Σis C linear operator

D : Ω0(Σ, E)→ Ω0,1(Σ, E)

which satisfies the Leibnitz rule

D(fξ) = fD(ξ) + (∂f)ξ, for ξ ∈ Ω0(Σ, E) and f ∈ Ω0(Σ)

Remark:

• A holomorphic structure on the bundle E → Σ is a maximal atlas of localcharts such that the transition functions gαβ : Uα ∩ Uβ → Gl(n,C) areholomorphic. Choosing a local chart U ⊂ Σ and a section s|U : U → E,

∂(gαβs) = ∂(gαβ)s + hαβ∂s = hαβ∂s, since gαβ are holomorphic. Its adeep theorem that every complex linear Cauchy-Riemann operator arisesin this way, hence there is a bijection


holomorphic structures onE → Σ ≡ Cauchy-Riemann operators onE → Σ

• A Hermitian connection on a vector bundle E is a C linear operator ∇ :Ω0(Σ, E)→ Ω1(Σ, E), such that ∇(fξ) = f∇(ξ)+(df)ξ clearly a Cauchy-Riemann operator defined by a connection is ∂ = (∇)0,1. The connectionis compatible with the hermitian metric 〈, 〉 on the bundle if

d〈ξ1, ξ2〉 = 〈∇ξ1, ξ2〉+ 〈ξ1,∇ξ2〉

for all f ∈ Ω0(Σ), ξ, ξ1, ξ2 ∈ Ω0(Σ, E)

5.2 Real linear Cauchy-Riemann operators

The space W k,p(E): We must first understand how to define Sobolev spaceof sections of a vector bundle. In general for any smooth vector bundle E → Σwe can define the space W k,p

loc (E) to consist of all sections whose expressions inall choices of local coordinates and trivializations are of class W k,p. When Σ iscompact define W k,p(E) := W k,p

loc (E). This space can be given the structure ofa Banach space.

Our main aim now is to study the Fredholm properties of spaces of these sec-tions and to state a infinite dimensional bundle version of the implicit functiontheorem. We are interested in the space of solutions to the non linear CauchyRiemann equation

u ∈ C∞(Σ,M)|Tu j = J Tu

which is best understood with the commutative diagram

Fig 5: The non linear Cauchy-Riemann equationDefinition 5.1:For any k ∈ N and p > 1 such that kp > 2 choose any smooth

connection on M and for any smooth map f ∈ C∞(Σ,M) choose a neighbour-hood Uf of the zero section in f∗TM then we define the space of W k,p smoothmaps from Σ→M byW k,p(Σ,M) = u ∈ C0(Σ,M)|u = expfη for any f ∈ C∞(Σ,M), η ∈W k,p(f∗TM), η(∂Σ) ⊂UfFor our purpose it will be imporant to consider the Banach manifold

Bk,p := W k,p(Σ,M)

with a Banach space bundle Ek−1,p → Bk,p whose fiber at u ∈ Bk,p is


Ek−1,pu := W k−1,p(Hom(TΣ, u∗TM))

As it turns out that Ek−1,p → Bk,p is a Banach space bundle such that

∂J : Bk,p → Ek−1,p sending u 7→ Tu+ J Tu j

where both the Riemann surface (Σ, j) and the almost complex manifold(M,J)are equipped with complex structures.

Remark: The zero set of ∂J is the set of solutions to the non linear Cauchy-Riemann equation. The topology of the space of solutions has no dependence

on k or p and to show ∂J−1

(0) has a nice structure we will need the implicitfunction theorem.

Definition 5.2 A bounded linear operator D : X → Y between Banachspaces is called Fredholm if both KerD and Y/ImD are finite dimensional.The latter space is called cokernel of D written as cokerD. The Fredholmindex of D is then defined as

ind(D) = dimker(D)− dimcoker(D)

The standard properties of fredholm operators areProposition 5.3: AssumeX and Y are Banach spaces and let Fred(X,Y ) ⊂

L(X,Y ) denote the space of Fredholm operators from X to Y

1. Fred(X,Y ) is an open subset of L(X,Y )

2. the map ind : Fred(X,Y )→ Z is continuous

3. if D ∈ Fred(X,Y ) is Fredholm and K ∈ L(X,Y ) is compact then D+Kis Fredholm

Obs: Note that the continuity of the map ind : Fred(X,Y ) → Z impliesthat the index function is locally constant, that is for any continuous family ofFredholm operators Dtt∈[0,1], ind(Dt) is constant

We are now in a position to state the Riemann-Roch theorem for a complexvector bundle over a compact Riemann surface with boundary although we willsolely be interested in the index formula supplied by the theorem

Theorem 5.4: Riemann-Roch theorem Let E → Σ be a complex vectorbundle over a compact Riemann surface with boundary. Let F ⊂ E|∂Σ be atotally real subbundle. Let D be a real linear Cauchy-Riemann operator on Ethen we have

1. the real Fredholm index of DF is given by

indDF = nχ(Σ) + µ(E,F )

where DF is an operator DF : W l,pF (Σ, E) → W l−1,p(Σ,Λ0,1T ∗Σ ⊗ E) ,

χ(Σ) is the Euler characterstics of the Riemann surface and µ(E,F ) is itsboundary Maslov index, n is the complex rank of E In case ∂Σ = φ wehave µ(E) = 2〈c1(E), [Σ]〉

2. If n = 1 then


• µ(E,F ) < 0⇒ DF is injective

• µ(E,F ) + 2χ(Σ) > 0⇒ DF is surjective

The Riemann Roch theorem gives explicit values for the dimension of thespace of sections when the associated Cauchy-Riemann operator is surjective asthe implicit function theorem states

Corollary 5.5: Implicit function theorem: If u ∈ ∂−1

J (0) and Du is

surjective then a neighbourhood of u in u ∈ ∂−1

J (0) admits the structure of asmooth finite dimensional manifold with

dim∂−1

J (0) = nχ(Σ) + 2〈c1(TM), [u]〉

5.3 Dimension of the space of pseudoholomorphic curves in an al-most complex manifold

Henceforth in this section we will consider (V, J) to be an almost complex man-ifold with boundary of real dimension 2n ≥ 4. We are interested in the space ofcompact embedded or immersed pseidoholomorphic curves in C ⊂ (V, J)Let us recall the notion of pseudoholomorphic curves. Consider a Riemann sur-face Σ. A pseudoholomorphic curve is a differentiable map f : Σ→ V such thatfor each σ ∈ Σ the derivative

dfσ : (TΣ)σ → (TV )f(σ)

is complex linear with respect to the complex structures. In case Σ = C andV = Cn the almost complex structure J : Cn → Cn can be parametrized by anopen set of matrices (µ) = µαβ then a pseudoholomorphic curve is given by thesolution of the nonlinear Cauchy-Riemann equation which now takes the formof

∂zα∂τ

+ Σβµαβ(z)∂zβ∂τ = 0

Let C0 be such a fixed pseudoholomorphic curve and assume first its boundaryis empty. Denote by ν to be the normal bundle of C0 and by Lν a Cauchy-Riemann operator asssociated with the normal bundle ν. Then by theorem 5.4we have

indLν = 1.χ(C0) + 2c1(ν) = 2(1− g + c1(ν))

using the fact that χ(C0) = 2(1− g) where g is the genus of the curve and when∂C0 = φ the boundary Maslov index is µ(ν) = 2c1(ν)

Our main theorem of this section isTheorem 5.6: If c1(ν) ≥ 2g − 1, then Lν is surjective, thus the space of pseu-doholomorphic curves near C0 is a manifold of dimension equal to the index ofLν i.e., 2(1− g + c1(ν))

Remark:

• Observe that the conclusion regarding the dimension of the space of pseu-doholomorphic curves is an immediate consequence of the implicit func-tion theorem (Corollary 5.5)


• For a sphere or disk g = 0 hence the chern class c1(ν) ≥ −1 in other wordsif the index indLν ≥ 0 then the space of holomorphic curves is a manifoldof dimension equal to the index

• For a curve C0 ⊂ V with boundary we require that ∂C0 lies in a totally realsurface with boundary denoted by ν∂ ⊂ ν ∩TW over ∂C0. The boundaryMaslov index for the curve is denoted by µ = µ(ν, ν∂) the formula for theindex of the Cauchy-Riemann operator now is

ind (Lν,∂) = 2(1− g) + (µ− σ)

where σ is the number of boundary components. For a disk σ = 1, g = 0hence index is µ+ 1

Theorem 5.7: If µ(ν, ν∂) ≥ 4g + 2σ − 3 then Lν,∂ is surjective. Thus thespace of holomorphic curves (C, ∂C) ⊂ (V,W ) near (C0, ∂C0) is a manifold ofdimension 2(1− g) + (µ− σ)

Corollary 5.8 If C0 is a disk with normal Maslov index µ ≥ −1 then Lν,∂is surjective. Thus the space of holomorphic curves near C is a manifold ofdimension µ+ 1.

We will atleast mention in the passing what it means to be a generalizedCauchy-Riemann operator over a holomorphic vector bundle for our case thenormal bundle over a Riemann surface

Definition 5.9: Let E be a holomorphic line bundle over a Riemann sur-face. A generalized ∂ operator is an operator of the form L = ∂ + a witha ∈ Ω0.1(EndRE)

Obs: Observe that with this definition L is not a Cauchy-Riemann operatorsince a is not complex linear but only real linear, however we have the followingresultFact: there exists b ∈ Lp(Ω0,1) such that ∂f + bf = 0

Proof: Simply define b for z ∈ Σ and v ∈ TzΣ as b(z)v = a(z)v.f(z)f(z) if f(z) 6= 0

and 0 otherwise

Theorem 5.10 : Let E be a holomorphic line bundle over a Riemann surfaceΣ and let L = ∂ + a be a generalized ∂ operator then

• if c1(E) < 0 , L is injective

• if c1(E) ≥ 2g − 1, L is surjective

Proof: Firstly observe that the space of holomorphic sections of the linebundle E is precisely Ker L. Since by definition the chern class can be writtenas

c1(E) = Σp∈Σνp(f).p


where f ∈ Ω0(Σ, E) is a holomorphic section and νp(f) being the order of thezero or pole of f at p ∈ Σ. Now the hypothesis c1(E) < 0 immediately forcesKer L = φ since νp(f) ≥ 0, f being holomorphic.for the second part observe that cokerL = h1(L) = h0(L∗ ⊗ KΣ) by Serreduality. Now the chern class

c1(L∗ ⊗KΣ) = −c1(E) + c1(KΣ) (c1(E∗) = −c1(E))

≤ −2g + 1 + 2g − 2 (c1(KΣ) = 2(g − 1))

≤ −1

⇒ h0(L∗ ⊗KΣ) = 0

Remark: By the adjoint arguement it can be shown that the first part ofthe theorem implies the second. Our last goal is to give a proof of theorem 5.7by a doubling arguement of the Riemann surface Σ

We recall here the definition of boundary Maslov index already encoun-tered in section 4.1. Since ∂Σ 6= φ recall lemma 4.5 by which the E admits atrivialization so the totally real subbundle F defines a continuous family of reallines in C. Let S1

j 1 ≤ j ≤ σ be the boundary components of Σ oriented so thatΣ is on the left. Then

Fj(θ) = fj(θ)R where fj(θ) : [0, 2π]→ S1

is such that fj(0) = +−fj(2π) then f2

j defines a loop in S2 and the Maslov indexis given by

µ(E,F ) := Σjdeg(f2j )

Remark:

• By the Claim following Corollary 4.6 we have already shown that thedefinition of µ(E,F ) is independent of the choice of the trivialization. Forexample (E,F ) = (TΣ, T∂Σ) then µ(E,F ) = 4− 4g − 2σ

Theorem 5.7 will follow from the following theoremTheorem 5.11: Let Σ, E, F, L be as above then

• if µ(E,F ) < 0, L is injective

• if µ(E,F ) ≥ 4g + 2σ − 3, L is surjective

Proof: Obtain the double of the Riemann surface Σ′

= Σ∪∂Σ Σ where Σ isthe mirror image of Σ and the corresponding points on the border are identified.then its genus g

′= 2g + σ − 1.

The doubling of the line bundle E is similar. Let E denote the line bundleover Σ with fiber Ez is Ez with the complex structure −i. Then the bundle E

′

over Σ′

is obtained by gluing E and E along ∂Σ. We have the following property

Property: The chern class c1(E′) = µ(E,F )

Proof: From the chern class axiom of theorem 4.3 we have

µ(E′, F ) = 2〈c1(E

′,Σ)〉

= 2µ(E,F ) (µ(E,F ) = µ(E,F ) =1

2µ(E

′, F ))

⇒ c1(E′, F ) = µ(E,F )

REFERENCES 30

We have now the doubled operator L′

: Γ(E′)→ Ω0,1(E

′) defined by

L′

|E = L and L′

|E = τ L τ .

We have the genus g(Σ′) = 2g+σ−1 and c1(E

′) = µ(E,F ) so that c1(E

′) ≥

2g′ − 1 is equivalent to µ(E,F ) ≥ 4g + 2σ − 3

Proposition 5.12 If L′

is injective respectively surjective so is LThe proof of theorem 5.11 is now immediate by applying theorem 5.10 and

keeing in mind proposition 5.12

Acknowledgement

I am deeply grateful to Dr. Jean Yves Welschinger for inviting me to InstituteCamille Jordan for the summer internship. The discussions with him regardingthe various aspects of Maslov index and related topics were stimulating andextremely motivating. I had also many nice discussions with Remi Cretois,PhD student of Dr. Jean Yves Welschinger. Apart from learning some excitingmathematics I had a pleasant stay in the beautiful city of Lyon. I would alsolike to thank Centre National de la Recherche Scientifique(CNRS) in Franceand the mathematics lab at Institute Camille Jordan for funding my visit.

References

[1] Joel Robbin and Dietmar Salamon, The Maslov index for paths TopologyVol 32 No. 4, 1992

[2] Paolo Piccione and Daniel Victor Tausk, On the geometry of Grassmanniansand the Symplectic group: the Maslov index and its applications Notes for ashort course given at the XI Escola de Geometria Diferencial UniversidadeFederal FLuminense, Niteroi, RJ Brazil

[3] V.I. Arnold, On charactersitic classes entering into conditions of quantiza-tion, Functional Analysis 1, 1967, 1− 8

[4] Simon K. Donaldson, What is a Pseudoholomorphic curve, Notices of theAMS, Volume 52, Number 9

[5] V.I. Arnold, The Sturm theorems and Symplectic geometry, Functional Anal-ysis 1, Vol 19, No. 4, 1985

[6] Semyon dyatlov, Hormander-Kashiwara and the Maslov indices, 2009

[7] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Inventionesmathematicae, Springer-Verlag, 1985

[8] Sabin Cautis, notes on Vector bundles on Riemann surfaces

[9] Chris Wendl, Institut fur Mathematik, Humboldt-Universitat at zu Berlin,Lectures on Holomorphic Curves in Symplectic and Contact Geometry

[10] Dusa Mcduff and Dietmar Salamon, J holomorphic Curves and SymplecticTopology, American Mathematical Society,Colloquium publications, Vol: 52

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[11] Dusa Mcduff and Dietmar Salamon, Introduction to Symplectic Topology,Oxford Mathematical Monographs.

[12] John Milnor, James D. Stasheff, Characteristic Classes, Annals of mathe-matical studies, Princeton University press

[13] Ivashkovich, S., Shevchishin, V, Deformations of non compact complexcurves and meromorphy of spheres, (Russian) Mat. Sb. 189(1998) No 9,23− 60

[14] Ivashkovich, S., Shevchishin, V, Pseduholomorphic curves and en-velopes of meromorphy of two spheres in CP 2, Preprint, Bochum(1995),ArXiv:math.CV/9804014

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