riemannian manifolds - syscon · 2017-03-17 · differentiating a vf along a curvecovariant...

Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection

Riemannian manifolds

Ravi N [email protected] 1

1Systems and Control Engineering,IIT Bombay, India

March 17, 2017

Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN


Outline

1 Differentiating a vector field along a curve

2 The covariant derivative

3 Affine connection

4 Riemannian connection



Differentiating a vector field along a curve

A curve in R3

• Let α(t) be a curve in R3 parametrized with time (t ∈ R.) LetFi(t); i = 1, 2, 3 be a basis parametrized with time .

• V is a vector field along α. So V (α(t)) ∈ Tα(t)R3.

• Let

V (t) =

3∑i=1

bi(t)Fi(t)

• Then

dV

dt=

3∑i=1

[dbi(t)

dtFi(t) + bi(t)

dFi(t)

dt]

• If the basis is constant, say Fi(t) = ∂∂xi, then

dV

dt=

3∑i=1

[dbi(t)

dt

∂

∂xi]



Frenet-Serre frames

Parametrizing with arclength

• Arclength s (an alternate parametrization)

s4=

∫ t

t0

⟨˙α(t), ˙α(t)

⟩0.5

dt

• The following equalities follow

ds

dt=⟨

˙α(t), ˙α(t)⟩0.5

dα(s(t))

dt=dα

ds

ds

dt⇒ dα

ds= [

dα

dt]/[ds

dt]

• Define

T (s)4=dα

dsunit tangent vector

•

‖T (s)‖ = 1⇒ d

ds〈T (s), T (s)〉 =

⟨2dT

ds, T

⟩= 0



Curvature and torsion

• Define k(s) =∥∥ dTds

∥∥ - the curvature.

• Define an orthonormal basis as

T (s), N(s), B(s)

where dTds

= k(s)N(s) and B(s) is a third, unit orthonormal vector thatpreserves orientation. (k(s) 6= 0.)

• Let F1 = T, F2 = N,F3 = B (the moving basis). Then

dT

ds= k(s)N(s)

dN

ds= −k(s)T + τ(s)B

dB

ds= −τ(s)N

• τ(s) denotes the torsion (a measure of the out-of-plane bending of thecurve.)



Outline



3 Affine connection




A metric on the tangent space

Notion of distanceThe need to define a notion of distance or the length of a curve on amanifold.

A metricA Riemannian metric on a differentiable manifold is a smoothly varyinginner-product on the tangent space G(x)(·, ·) (also termed a covarianttwo-tensor), that satisfies the following properties. For each x ∈M ,G(x)(·, ·) : TxM × TxM→R satisfies

• G(x)(v, v) ≥ 0 ∀v ∈ TxM and G(x)(v, v) = 0 iff v = 0. Positivedefinite

• G(x)(v, w) = gx(w, v) ∀v, w ∈ TxM Symmetric

• G(x)(α1v1 + α2v2, w) = α1G(x)(v1, w) + α2G(x)(v2, w) ∀v1, v2, w ∈TxM, ∀α1, α2 ∈ R Linearity (in fact bilinear)

A smooth manifold endowed with a Riemannian metric is a Riemannianmanifold.



Length of a curve

Let M be a Riemannian manifold and α(·) be a smooth curve from[a, b] ⊂ I→M . Then the length of the curve is defined as

L(α)4=

∫ b

a

√G(α(t))(

dα

dt,dα

dt)dt



Examples of Riemannian metrics

Rn

The Riemannian metric on Rn is the familiar Euclidean metric.

G(x)(v, w) = vT Iw where I is the n× n identity matrix.




Sphere S2Coordinates (θ, φ) ∈ ((0, π), [0, 2π]).

Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, φ) = (sin θ cosφ, sin θ sinφ, cos θ)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3

vθ∂

∂θ+ vφ

∂

∂φ→[Df(θ, φ)]

[vθvφ

]= [(cos θ cosφ)vθ − (sin θ sinφ)vφ]

∂

∂x+ [(cos θ sinφ)vθ + (sin θ cosφ)vφ]

∂

∂y

−(sin θ)vθ∂

∂z

G(θ, φ)(v, w) =[vθ vφ

] [ 1 00 sin2 θ

] [wθwφ

]




Sphere S2Coordinates (θ, φ) ∈ ((0, π), [0, 2π]).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, φ) = (sin θ cosφ, sin θ sinφ, cos θ)

Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3

vθ∂

∂θ+ vφ

∂

∂φ→[Df(θ, φ)]

[vθvφ


∂


∂

∂y

−(sin θ)vθ∂

∂z


] [ 1 00 sin2 θ

] [wθwφ

]




Sphere S2Coordinates (θ, φ) ∈ ((0, π), [0, 2π]).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, φ) = (sin θ cosφ, sin θ sinφ, cos θ)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3

vθ∂

∂θ+ vφ

∂

∂φ→[Df(θ, φ)]

[vθvφ


∂


∂

∂y

−(sin θ)vθ∂

∂z


] [ 1 00 sin2 θ

] [wθwφ

]




Cylinder S1 ×R1

Coordinates (θ, z) ∈ ([0, 2π), (−∞,∞)).

Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, z) = (cos θ, sin θ, z)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3

vθ∂

∂θ+ vz

∂

∂z→[Df(θ, φ)]

[vθvφ

]= [− sin θvθ]

∂

∂x+ [cos θvθ]

∂

∂y+ vz

∂

∂z

G(θ, φ)(v, w) =[vθ vz

] [ 1 00 1

] [wθwz

]




Cylinder S1 ×R1

Coordinates (θ, z) ∈ ([0, 2π), (−∞,∞)).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, z) = (cos θ, sin θ, z)

Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3

vθ∂

∂θ+ vz

∂

∂z→[Df(θ, φ)]

[vθvφ

]= [− sin θvθ]

∂

∂x+ [cos θvθ]

∂

∂y+ vz

∂

∂z


] [ 1 00 1

] [wθwz

]




Cylinder S1 ×R1

Coordinates (θ, z) ∈ ([0, 2π), (−∞,∞)).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, z) = (cos θ, sin θ, z)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3

vθ∂

∂θ+ vz

∂

∂z→[Df(θ, φ)]

[vθvφ

]= [− sin θvθ]

∂

∂x+ [cos θvθ]

∂

∂y+ vz

∂

∂z


] [ 1 00 1

] [wθwz

]




The Lie group SO(3)

The Riemannian metric on SO(3) is induced by the inner product on theLie algebra so(3), wherein

G(R)(vR, wR)4=⟨

Ω1, Ω2

⟩so(3)

= −1

2Trace(Ω1Ω2) (1)

where vR = RΩ1 ∈ TR SO(3) and wR = RΩ2 ∈ TR SO(3), the lefttranslations by R of Ω1 and Ω2, respectively.Here Ω1 and Ω2 belong to theLie algebra so(3). Note that we have used the left-invariant property of thevector field on SO(3).



Simple mechanical systems and Riemannian manifolds

For a simple mechanical system, the kinetic energy is a Riemannian metric .

Minimizing the KE for a mechanical system between any two fixed intialand final points is the equivalent to finding a shortest path or a geodesicbetween two points on a Riemannian manifold with the metric being thekinetic energy.



Riemannian manifolds as submanifolds of Rn

• Suppose M is a Riemannian manifold of dimension m immersed in Rn(n ≥ m.)

• Let α(·) be a smooth curve from [a, b](⊂ R)→M . Let V (α(t)) be asmooth vector field defined on the curve α(·). This means, for every t

V (t) ∈ Tα(t)M

• Question What is dVdt

?

• Though V (t) ∈ Tα(t)M , dVdt

need not be in Tα(t)M .

• If we now look at the point α(t) in Rn, the tangent space to Rn at anypoint is Rn and is hence n-dimensional.

• However, we are concerned about the rate of change of V (t) along themanifold. Hence we define a new quantity called the covariantderivative

DV

dt

4= π(

dV

dt) where π denotes the projection on Tα(t)M .



Properties of the covariant derivative

The covariant derivative satisfies

•D

dt(V +W ) =

DV

dt+DW

dt•

D(fV )

dt=df

dtV + f

DV

dt•

d(g(V,W ))

dt= g(

DV

dt,W ) + g(V,

DW

dt)



Formulae for the covariant derivative

• Let u4= (u1, . . . , um) be local coordinates for M around a point φ(p),

where φ−1 is a parametrization

φ−1 : Rm→Rn x = φ−1(u) = (g1(u), . . . , gn(u))

• The associated tangent map is

φ−1∗ : TM→TRn

• The basis ∂∂u1 , . . . ,

∂∂um for the tangent space to M at φ(p) in the

coordinates u, can be expressed in terms of the basis ∂∂x1

, . . . , ∂∂xn

using the tangent map as

Fjp4= (φ−1

∗ )φ(p)∂

∂uj=

n∑k=1

[∂gk

∂uj]φ(p)

∂

∂xk



Computing DFkdt

• Let α be a curve and V be a vector field defined along α.

• Using the basis F1(t), · · · , Fm(t), we express V (t) =∑mk=1 b

k(t)Fk(t),where bk(t) is a smooth function.

• Now

DV (t)

dt=

m∑k=1

dbk

dtFk(t) + bk(t)

DFkdt

• Our interest is to compute DFkdt

.



Computing DFidt

• Recall

Fjp4= (φ−1

∗ )φ(p)∂

∂uj=

n∑k=1

[∂gk

∂uj]φ(p)

∂

∂xk

• ThendFidt

=

n∑k=1

m∑l=1

[∂2gk

∂ul∂ui]φ(p)

dul

dt

∂

∂xk

• Taking the projection π,

DFidt

= πp(dFidt

) =n∑k=1

m∑l=1

[∂2gk

∂ul∂ui]φ(p)

dul

dtπp[

∂

∂xk]

• Express

πp[∂

∂xk] =

m∑j=1

ajkFj

• Define

Γjli4=

n∑k=1

[∂2gk

∂ul∂ui]φ(p)a

jk



The Christoffel symbols

• Then

DFidt

=m∑j=1

m∑l=1

Γjlidul

dtFj

• The covariant derivative

DV (t)

dt=

m∑k=1

dbk

dtFk(t) + bk(t)

m∑j=1

m∑l=1

Γjlkdul

dtFj

=m∑k=1

[dbk

dt+

m∑j=1

m∑l=1

Γkljdul

dtbj(t)]Fk

• The Christoffel symbol Γklj denotes the kth component of the covariantderivative of Fl along the curve in which only the jth coordinate isallowed to vary.



Computing Christoffel symbols

• Compute the Christoffel symbols from the Riemannian metric as

Γkij =1

2Gkl(∂Gil

∂xj+∂Gjl∂xi

− ∂Gij∂xl

)

where Gkl stands for the inverse of .

•

∇ ∂∂xi

∂

∂xj= Γkij

∂

∂xk

Γkij are the n3 Christoffel symbols for ∇ in the specified coordinates.



A vector field along another vector field

Covariant derivative of Y with respect to X

∇XY = (∂Y k

∂xiXi + ΓkijX

iY j)∂

∂xk

Covariant derivative of Y along a curve γ

∇γ(t)Y (t) = (Y k(t) + Γkij(γ(t))xi(t)Y j(t))∂

∂xk

t→(x1(t), . . . , xn(t)) is a local representation of γ.



Outline



3 Affine connection




An affine connection

General Riemannian manifoldsHow do we introduce the notion of a covariant derivative in a more generalRiemannian manifold (not necessarily submersed in Rn ?) We need thenotion of an affine connection.

The notion of a connection provides a tool for differentiating vectors alongcurves; in particular, we can talk of the acceleration of a curve in M .

DefinitionAn affine connection ∇ on a differentiable manifold is a mapping

∇ : X (M)×X (M)→X (M)

denoted by (X,Y )→∇XY , which satisfies the following properties

• ∇fX+hY Z = f∇XZ + h∇Y Z• ∇X(Y + Z) = ∇X(Y ) +∇X(Z)

• ∇X(fY ) = f∇X(Y ) +X(f)Y



Covariant derivative

The connection helps us to define the derivative of a vector field alongcurves called the covariant derivative.

DefinitionLet M be a differentiable manifold with an affine connection ∇. Thereexists a unique corespondence which associates to a vector field V along adifferentiable curve α : I→M , another vector field DV

dtalong α, called the

covariant derivative of V along α, such that

• Ddt

(V +W ) = DVdt

DWdt

• D(fV )dt

= dfdtV + f DV

dt

• If V is induced by a vector field Y ∈ X (M), say V (t) = Y (α(t)), then

DV

dt= ∇ dα

dtY

Unlike the Lie derivative, the covariant derivative brings in additionalstructure to the manifold and is not instrinsic to the manifold itself.



Parallelism

DefinitionLet M be a differentiable manifold with an affine connection ∇. A vectorfield V along a curve α : I→M is called parallel when DV

dt= 0, for all t ∈ I.

GeodesicsLet M be a differentiable manifold with an affine connection ∇. A curveα : I→M is called a geodesic if ∇ dα

dt

dαdt

= 0 for all t ∈ I.

Motion in R3

Look at the motion of a point mass in R3. Call its trajectory α(t). Then itsvelocity vector is dα

dt. For what type of α is

D

dt(dα

dt) = 0 ?

Straight lines !. α(t) = k1t+ k2.



Outline



3 Affine connection




Metric compatible connection

DefinitionA connection on a Riemannian manifold is said to be compatible with themetric g(·, ·), when for any smooth curve α and any pair of parallel vectorfields V and W (i. e. DV

dt= 0, DW

dt= 0) along α, we have

gα(t)(V (α(t)),W (α(t))) = constant

TheoremA connection ∇ on a Riemannian manifold is said to be compatible with themetric g(·, ·) if and only if

X(g(Y,Z)) = g(∇XY,Z) + g(Y,∇XZ)



Riemannian connection

DefinitionA connection ∇ on a smooth manifold is said to be symmetric when

∇XY −∇YX = [X,Y ]

Riemannian connection

TheoremThere exists a unique affine connection ∇ on a Riemannian manifold Mthat is both symmetric and is compatible with the Riemannian metric. Thisis called the Levi-Civita (or Riemannian) connection.


riemannian manifolds - syscon · 2017-03-17 · differentiating a vf along a curvecovariant...

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