Riemannian Geometry, continued - Imageimage.diku.dk › MLLab › SummerSchools ›...

U N I V E R S I T Y O F C O P E N H A G E N U N I V E R S I T Y O F C O P E N H A G E N

Faculty of Science

Riemannian Geometry, continued

François LauzeDepartment of Computer ScienceUniversity of Copenhagen

Information Geometry SchoolSlide 1/68

Outline

1 Introduction

2 Lie DerivativesLie Derivatives of FunctionsLie Brackets

3 Riemannian ConceptsMetricGradient FieldLengths and Distance

4 ConnectionsWhy connectionsAffine ConnectionsParallelism

5 Riemannian GeometryLevi-Civita ConnectionsRiemannian GeodesicsExponential and Log MapsJacobi Fields and Curvature

/68 — François Lauze — Differential Geometry — September 2014,

Optimization

• f : U ⊂ Rn → R differentiable. Search for min point

• Taylor Series:f (x + h) = f (x) +∇x · h + o(|h|)

• From it, Gradient descent

xn+1 = xn − τ∇xf

• Also Newton and variants

xn+1 = xn − Hxf−1∇xf

Optimization

• f : U ⊂ Rn → R differentiable. Search for min point• Taylor Series:

f (x + h) = f (x) +∇x · h + o(|h|)

Optimization

f (x + h) = f (x) +∇x · h + o(|h|)

Optimization

f (x + h) = f (x) +∇x · h + o(|h|)

Optimization on Manifolds

M manifold, f :M→ R function to optimize. Problems• Addition of point and vector not defined!

• Gradient needs an inner product.

• Set of vector tangent toM at x : TxM. Set of vector tangent toMaty : TxM.

x 6= y TxM 6= TyM

• Hessian as differential of gradient: how to differentiate vector fields?

x 6= y TxM 6= TyM

Outline

1 Introduction

Directional Derivatives of a Function

• f :M→ R, p ∈M, X vector field onM:

(Xf )p := dpf X (p).

• Computes the image of X (p) by linear mapping dpf .• In coordinates:

f (p) = f (x1(p), . . . , xn(p)), X (p) =n∑

X i (p)∂

(Xf )p =n∑

X i (p)∂f∂xi

• Called Lie Derivative of f along X .

• Computes the image of X (p) by linear mapping dpf .

• In coordinates:

f (p) = f (x1(p), . . . , xn(p)), X (p) =n∑

X i (p)∂

(Xf )p =n∑

X i (p)∂f∂xi

f (p) = f (x1(p), . . . , xn(p)), X (p) =n∑

X i (p)∂

(Xf )p =n∑

X i (p)∂f∂xi

f (p) = f (x1(p), . . . , xn(p)), X (p) =n∑

X i (p)∂

(Xf )p =n∑

X i (p)∂f∂xi

Outline

1 Introduction

Flows of Vector Fields

• Integral curves of X:

cX (p, t),∂cX (p, t)

∂t= X (cX (p, t)), c(p, 0) = p.

• Flows ϕXt : p :7→ Cx (p, t) diffeomorphisms (t small)

∂t= X (cX (p, t)), c(p, 0) = p.

• Flows ϕXt : p :7→ Cx (p, t) diffeomorphisms (t small)

∂t= X (cX (p, t)), c(p, 0) = p.

• Flows ϕXt : p : 7→ Cx (p, t) diffeomorphisms (t small)

Lie Brackets

• dpϕXt : TPM→ TϕXt (p)M invertible: reverse time!• Differentiate field Y along field X at p:

[X ,Y ]p = limt→0

(dpϕXt )−1 Y (ϕXt (p))− Y (p)

• In coordinates: X =∑

X i∂xi , Y =∑i

Y ∂xi

[X ,Y ] =∑

X j ∂Y i

∂xj− Y j ∂X i

Lie Brackets of Coordinate Systems

• Given a coordinate system: special vector fields ∂xi : images of chartbasis vectors ~ei .

• Satisfy[∂xi , ∂xj ] = 0.

• Their components in vector field basis ∂x1 , . . . , ∂xn are constant!

Outline

1 Introduction

Riemannian Metric

• Smooth family of inner products on each tangent space.

• P ∈M, gP inner product on TPM, i.e. positive definite bilinear form.

v ,w ∈ TPM 7→ gP(v ,w) ∈ R.

• Smooth: if V , W smooth vector fields onM,

P 7→ gP(V (P),W (P)) smooth functionM→ R.

Riemannian Metric

v ,w ∈ TPM 7→ gP(v ,w) ∈ R.

Riemannian Metric

v ,w ∈ TPM 7→ gP(v ,w) ∈ R.

Riemannian Metric: Embedded Manifolds

• Metric from the ambient space: S2 ⊂ R3, each TPS2 is a subspace of R3,restrict the usual inner product to it.

• What is TPS2

Example: Sphere and Stereographic Projection

ΦN : S2\N → R2

(x , y , z) 7→(

x1− z

1− z

) ϕN : R2 → S2

(u, v) 7→ 1`

(2u, 2v , `− 2)

(` = u2 + v2 + 1)

Metric of the sphere via stereographic projection?

ΦN : S2\N → R2

(x , y , z) 7→(

x1− z

1− z

ϕN : R2 → S2

(u, v) 7→ 1`

(2u, 2v , `− 2)

(` = u2 + v2 + 1)

ΦN : S2\N → R2

(x , y , z) 7→(

x1− z

1− z

) ϕN : R2 → S2

(u, v) 7→ 1`

(2u, 2v , `− 2)

(` = u2 + v2 + 1)

ΦN : S2\N → R2

(x , y , z) 7→(

x1− z

1− z

) ϕN : R2 → S2

(u, v) 7→ 1`

(2u, 2v , `− 2)

(` = u2 + v2 + 1)

• Mapping between T(u,v)R2 and TϕN (u,v)S2: JϕN (u,v)

JϕN (u,v) =2`2

`− 2u2 −2uv−2uv `− 2v2

• e1 = (1, 0)> and e2 = (0, 1)> basis of T(u,v)R2 = R2.

∂uϕN (u,v) = JϕN (u,v)e1 =2`2

`− 2u2

−2uv2u

, ∂v =2`2

−2uv`− 2v2

• ∂u and ∂v tangent vectors to TϕN (u,v)S2 ⇐⇒ ∂u⊥ϕN(u, v), ∂v⊥ϕN(u, v):check it!

JϕN (u,v) =2`2

`− 2u2 −2uv−2uv `− 2v2

∂uϕN (u,v) = JϕN (u,v)e1 =2`2

`− 2u2

−2uv2u

, ∂v =2`2

−2uv`− 2v2

JϕN (u,v) =2`2

`− 2u2 −2uv−2uv `− 2v2

∂uϕN (u,v) = JϕN (u,v)e1 =2`2

`− 2u2

−2uv2u

, ∂v =2`2

−2uv`− 2v2

• Compute inner products

〈∂u, ∂u〉R3 = 〈∂v , ∂v 〉R3 =4`2

〈∂u, ∂v 〉R3 = 0.

• Corresponding family of inner products on R2:

g(u,v) =4

(u2 + v2 + 1)2

(1 00 1

• Also written in term of “squared-length-element”

ds2 =4

(u2 + v2 + 1)2

(du2 + dv2

〈∂u, ∂u〉R3 = 〈∂v , ∂v 〉R3 =4`2

〈∂u, ∂v 〉R3 = 0.

g(u,v) =4

(u2 + v2 + 1)2

(1 00 1

ds2 =4

(u2 + v2 + 1)2

(du2 + dv2

〈∂u, ∂u〉R3 = 〈∂v , ∂v 〉R3 =4`2

〈∂u, ∂v 〉R3 = 0.

g(u,v) =4

(u2 + v2 + 1)2

(1 00 1

ds2 =4

(u2 + v2 + 1)2

(du2 + dv2

Riemannian Metric: Charts / Parametrisations

• With local parametrization θ(x) = (x1, . . . , xn)→ M, smooth family ofpositive definite matrices:

gx11 . . . gx1n...

...gxn1 . . . gxnn

• u =∑n

i=1 ui∂xi , v =∑n

i=1 vi∂xi

〈u, v〉x = (u1, . . . , un)gx(v1, . . . , vn)t

Riemannian Metric: Charts / Parametrisations

• With local parametrization θ(x) = (x1, . . . , xn)→ M, smooth family ofpositive definite matrices:

gx11 . . . gx1n...

...gxn1 . . . gxnn

• u =∑n

i=1 ui∂xi , v =∑n

i=1 vi∂xi

〈u, v〉x = (u1, . . . , un)gx(v1, . . . , vn)t

Riemannian Metric with Charts

Orthonormal vectors on TPM not transported to standard orthonormal oneson parameter space. Need to locally adapt the metric / inner product.

With Charts: Fisher Information Metric

• 1D Gaussian distributions: fµ,σ(x) = 1√2πσ2 e−

(x−µ)2

• Parametrized by (θ1, θ2) = (µ, σ) ∈ E = R× R++

• Fisher Information Metric on E :(g(µ,σ)

∫ ∞−∞

∂log fµ,σ(x)

∂θi

∂log fµ,σ(x)

∂θj, fµ,σ(x) dx

• (Stefan?)

gµ,σ =1σ2

(1 00 2

), ds2

F =dµ2 + 2dσ2

(x−µ)2

∫ ∞−∞

∂log fµ,σ(x)

∂θi

∂log fµ,σ(x)

• (Stefan?)

gµ,σ =1σ2

(1 00 2

), ds2

F =dµ2 + 2dσ2

(x−µ)2

∫ ∞−∞

∂log fµ,σ(x)

∂θi

∂log fµ,σ(x)

• (Stefan?)

gµ,σ =1σ2

(1 00 2

), ds2

F =dµ2 + 2dσ2

(x−µ)2

∫ ∞−∞

∂log fµ,σ(x)

∂θi

∂log fµ,σ(x)

• (Stefan?)

gµ,σ =1σ2

(1 00 2

), ds2

F =dµ2 + 2dσ2

Riemannian Manifold

A differential manifold with a Riemannian metric is a Riemannian manifold.

Outline

1 Introduction

Gradient, Gradient vector field.

• M Riemannian, f : M → R differentiable.

dP f (h) = 〈v , h〉P , for a unique v .

• v := ∇fP is the gradient of f at P.

• P 7→ ∇fP is the gradient vector field of f.

• One can thus make gradient descent/ascent... Not possible withoutRiemannian structure.

• Gradient of a spherical function f : S2 → R expressed in stereographicparametrization (easy)?

Outline

1 Introduction

• Curve c : [a, b]→ M, M Riemannian. For all t , c′(t) ∈ Tc(t)M is thevelocity of c at time t .

• It has length ‖c(t)‖c(t) =√〈c′(t), c′(t)〉c(t)

• Define the length of c as

`(c) =

a‖c(t)‖c(t) dt

as in the Euclidean case, by now with variable inner products.

`(c) =

a‖c(t)‖c(t) dt

`(c) =

a‖c(t)‖c(t) dt

`(c) =

a‖c(t)‖c(t) dt

Example

Blue curve:

c =( 1

2 t), t ∈ [−1, 1].

Red curve d : image of the blue!

`(d) =

√u(t)2 + v(t)2 dt

(u(t)2 + v(t)2 + 1)2 =

dt( 14 + t2 + 1

)2 =89

(sympy...)

Geodesic Distances, Riemannian Geodesics

• (M, 〈, 〉) distance between two points P and Q: inf of length of curvesjoining P to Q

d(P,Q) = infc,c(0)=P,c(1)=Q

0|c(t)| dt

• It is a metric distance: a bit of work to show thatd(P,Q) = 0 ⇐⇒ P = Q (the rest is obvious).

• A curve with min distance is a Riemannian Geodesic.

Shortest curve: red curve, arcof great circle: a Riemanniangeodesic.

0|c(t)| dt

Outline

1 Introduction

Acceleration of a Curve

• In Rn

c(0) = limt→0

c(t)− c(0)

• On a manifoldM: c(t) ∈ Tc(t)M 6= Tc(0)M 3 c(0)

• Need for a “device” that “connects” tangent spaces of close enoughpoints.

• Such a device is provided by an affine connection.

• In Rn

c(0) = limt→0

c(t)− c(0)

• In Rn

c(0) = limt→0

c(t)− c(0)

• In Rn

c(0) = limt→0

c(t)− c(0)

Lie Brackets?

• The Lie Bracket [X ,Y ]: Can transport Y along flow lines of X .

• [X ,Y ]p depends on values of X around p, not just X (p), from classicalrelation

[fX ,Y ]p = −dpf Y (p) + f (p)[X ,Y ]p

• choose f s.t. f (p) = 1, dpf 6= 0:

f (p)X (p) = X (p), f [X ,Y ]p 6= [X ,Y ]p!

• We want a “device” that depends only of X (p) at p!

Lie Brackets?

[fX ,Y ]p = −dpf Y (p) + f (p)[X ,Y ]p

• choose f s.t. f (p) = 1, dpf 6= 0:

f (p)X (p) = X (p), f [X ,Y ]p 6= [X ,Y ]p!

Lie Brackets?

[fX ,Y ]p = −dpf Y (p) + f (p)[X ,Y ]p

• choose f s.t. f (p) = 1, dpf 6= 0:

f (p)X (p) = X (p), f [X ,Y ]p 6= [X ,Y ]p!

Lie Brackets?

[fX ,Y ]p = −dpf Y (p) + f (p)[X ,Y ]p

• choose f s.t. f (p) = 1, dpf 6= 0:

f (p)X (p) = X (p), f [X ,Y ]p 6= [X ,Y ]p!

Outline

1 Introduction

Affine Connection

An affine connection on the manifoldM is a mapping of vector fields X andYto a vector field ∇X Y such that

1 R-linearity in Y : ∇X (Y + λY ′) = ∇X Y + λ∇X Y ′, λ ∈ R,

2 C∞(M)-linearity in X : ∇X+f X ′Y = ∇X Y + f∇X ′Y , f :M→ R smooth,

3 Leibniz-rule ∇X (f Y ) = (Xf ) Y + f ∇X Y , f :M→ R smooth.

• Point 1: usual linearity of differential operators.

• Point 3: generalization of the product rule (f g)′ = f ′g + f g′.

• Point 2: No dpf in 2: this makes (∇X Y )p dependent only on X (p) (and Yaround p).

Affine Connection

Simple Example – I

• Seems complicated? Simplest example: vector fields on Rn,X ,Y 7→ ∇X Y = JY (X ), JY Jacobian of Y .

• Exercise: check points 1, 2, and 3.

• Satisfy ∇X Y −∇Y X = JY (X )− JX (Y ) = [X ,Y ].

(Such a connection is symmetric).

Christoffel Symbols

• Expansion of ∇ in a coordinate system (x1, . . . , xn):

X =n∑

X i∂xi , Y =n∑

Y i∂xi

∇X Y =n∑

X in∑

∇∂xiY j∂xj Points 1 and 2

X in∑

(∂Y j

∂xi∂xj + Y j∇∂xi

)Leibniz rule

• ∇∂xi∂xj vector field: write as

∇∂xi∂xj =

n∑k=1

Γkij∂xk

• Γkij : n3 functions: Christoffel symbols of ∇ (for the given

parametrization).

Christoffel Symbols

X =n∑

X i∂xi , Y =n∑

Y i∂xi

∇X Y =n∑

X in∑

(∂Y j

)Leibniz rule

∇∂xi∂xj =

n∑k=1

Γkij∂xk

parametrization).

Christoffel Symbols

X =n∑

X i∂xi , Y =n∑

Y i∂xi

∇X Y =n∑

X in∑

(∂Y j

)Leibniz rule

∇∂xi∂xj =

n∑k=1

Γkij∂xk

parametrization).

Christoffel Symbols

X =n∑

X i∂xi , Y =n∑

Y i∂xi

∇X Y =n∑

X in∑

(∂Y j

)Leibniz rule

∇∂xi∂xj =

n∑k=1

Γkij∂xk

parametrization).

Christoffel Symbols

X =n∑

X i∂xi , Y =n∑

Y i∂xi

∇X Y =n∑

X in∑

(∂Y j

)Leibniz rule

∇∂xi∂xj =

n∑k=1

Γkij∂xk

parametrization).

Christoffel Symbols

• (Very simple) exercise Compute the Christoffel symbols of theconnection on Rn: ∇X Y = JY (X ).

• ∂xi = ei constant vector field! Jei ≡ 0.

• ∇∂xi∂xj = Jej (ei ) = 0 =

∑nk=1 Γk

ij∂xk : Γkij ≡ 0.

Christoffel Symbols

• ∇∂xi∂xj = Jej (ei ) = 0 =

∑nk=1 Γk

Christoffel Symbols

• ∇∂xi∂xj = Jej (ei ) = 0 =

∑nk=1 Γk

Covariant Derivative – Acceleration

• Covariant Derivative: differentiation of vector field X along a curveγ(t) ∈M

X (t) = X (t) ∈ Tγ(t)M

• From a connection (with slight abuse...) γ(t) velocity vector field of γ

X (t) := (∇γX )γ(t)

• (Covariant) acceleration of γ:

γ(t) :=Ddtγ(t) = (∇γ γ)γ(t)

X (t) = X (t) ∈ Tγ(t)M

X (t) := (∇γX )γ(t)

X (t) = X (t) ∈ Tγ(t)M

X (t) := (∇γX )γ(t)

Covariant Derivatives and Christoffel Symbols

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

dt∂xj + v j∇∑n

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i

dt∂xi v(t) =

n∑i=1

v i (t)∂xi

v =n∑

(v j∂xj

dt∂xj + v j D

dt∂xj

i=1dxidt ∂xi

dt∂xj +

n∑i=1

v j dx i

dt∇∂xi

n∑i,j=1

v j dx i

Simplest Example – II

• Covariant derivative associated to the (trivial) connection ∇X Y = JY (X )on Rn:

f (γ(t)) = Jγ(t)f γ(t) =df (γ(t))

• Christoffel symbols are ≡ 0.

f (γ(t)) = Jγ(t)f γ(t) =df (γ(t))

A Concrete Construction

• M ⊂ Rn. A vector field X on M can be seen as a vector field on Rn and acurve γ on M can be seen as a curve in Rn. Then

1 Compute the usual derivative

˜X(t) =ddt

X(γ(t))

its a vector field on Rn but not a tangent vector field on M in general.2 Project ˜X(t) orthogonally on Tγ(t)M ⊂ R3. The result is a covariant

derivative!

Outline

1 Introduction

Parallelism

• X vector field along γ is parallel if its covariant derivative is 0

• the X (t) are parallel-transported along γ.

• A curve is geodesic or autoparallel if its velocity is parallel-transportedalong itself.

γ(t) =Dγ(t)

dt= ∇γ(t)γ(t) = 0.

• γ is the covariant acceleration.

Parallelism

γ(t) =Dγ(t)

dt= ∇γ(t)γ(t) = 0.

Parallelism

γ(t) =Dγ(t)

dt= ∇γ(t)γ(t) = 0.

Parallelism

γ(t) =Dγ(t)

dt= ∇γ(t)γ(t) = 0.

Parallel Transport ODEs

• Parallel Transport Equation:dv1

dt +∑n

i,j=1 v j dx i

dt Γ1ij = 0

...dvn

dt +∑n

i,j=1 v j dx i

dt Γnij = 0

Linear system. Initial value problem always has solutions. If Γijk ≡ 0:solutions are constant!

• Geodesic Equations: replace v by dxdt :

dt2 +∑n

i,j=1dx j

dtdx i

dt Γ1ij = 0

...d2xn

dt2 +∑n

i,j=1dx j

dtdx i

dt Γnij = 0

Second order system. Initial values problem always has solutions. IfΓijk ≡ 0: solutions are straight lines!

Outline

1 Introduction

Metric Connections

• A connection onM is compatible with the Riemannian metric 〈, 〉 if

X 〈Y ,Z 〉 :=(d 〈Y ,Z 〉

)X = 〈∇X Y ,Z 〉 + 〈Y ,∇Z Y 〉

• With covariant derivatives:

ddt〈Y (t),Z (t)〉 =

⟨DY (t)

dt,Z (t)

⟨Y (t),

DZ (t)dt

⟩• Generalizes usual rule on Rn:

ddt〈f (t), g(t)〉 =

⟨df (t)

dt, g(t)

⟨f (t),

dg(t)dt

⟩Leibniz product rule extended to inner product.

Metric Connections

X 〈Y ,Z 〉 :=(d 〈Y ,Z 〉

)X = 〈∇X Y ,Z 〉 + 〈Y ,∇Z Y 〉

ddt〈Y (t),Z (t)〉 =

⟨DY (t)

dt,Z (t)

⟨Y (t),

DZ (t)dt

• Generalizes usual rule on Rn:

ddt〈f (t), g(t)〉 =

⟨df (t)

dt, g(t)

⟨f (t),

dg(t)dt

Metric Connections

X 〈Y ,Z 〉 :=(d 〈Y ,Z 〉

)X = 〈∇X Y ,Z 〉 + 〈Y ,∇Z Y 〉

ddt〈Y (t),Z (t)〉 =

⟨DY (t)

dt,Z (t)

⟨Y (t),

DZ (t)dt

⟩• Generalizes usual rule on Rn:

ddt〈f (t), g(t)〉 =

⟨df (t)

dt, g(t)

⟨f (t),

dg(t)dt

Deus Ex Machina: Levi-Civita Connections

• 〈, 〉 Riemannian metric on manifoldM. There exists a uniqueconnection ∇ onM such that

1 ∇ is compatible with the metric,

2 ∇ is symmetric: ∇X Y −∇Y X − [X ,Y ] = 0

• Levi-Civita connection associated to the metric.

• Expressed in terms of the metric and Lie brackets

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

Simplest Example – III

• Our connection on Rn : ∇X Y = JY (X ).

• We already saw it is symmetric!

• It is metric for the Euclidean metric!

X 〈Y ,Z 〉 := x 7→(dx 〈Y ,Z 〉

)(X (x))

• Leibniz for inner product:(dx 〈Y ,Z 〉

)(X (x)) = 〈JxY (X (x)),Z (x)〉 + 〈Y (x), JxZ (X (x))〉

= 〈JY (X ),Z 〉 (x) + 〈Y , JZ (X )〉 (x)

• It is the Levi-Civita connection for the Euclidean metric of Rn.

X 〈Y ,Z 〉 := x 7→(dx 〈Y ,Z 〉

)(X (x))

= 〈JY (X ),Z 〉 (x) + 〈Y , JZ (X )〉 (x)

X 〈Y ,Z 〉 := x 7→(dx 〈Y ,Z 〉

)(X (x))

= 〈JY (X ),Z 〉 (x) + 〈Y , JZ (X )〉 (x)

X 〈Y ,Z 〉 := x 7→(dx 〈Y ,Z 〉

)(X (x))

= 〈JY (X ),Z 〉 (x) + 〈Y , JZ (X )〉 (x)

X 〈Y ,Z 〉 := x 7→(dx 〈Y ,Z 〉

)(X (x))

= 〈JY (X ),Z 〉 (x) + 〈Y , JZ (X )〉 (x)

Concrete Construction - II• M ⊂ Rn. A vector field X on M can be seen as a vector field on Rn and a

curve γ on M can be seen as a curve in Rn. Then

1 Compute the usual derivative

˜X(t) =ddt

X(γ(t))

its a vector field on Rn but not a tangent vector field on M in general.2 Project ˜X(t) orthogonally on Tγ(t)M ⊂ R3. The result is the covariant

derivative associated with the Levi-Civita Connection (for induced innerproduct)!

curve γ on M can be seen as a curve in Rn. Then1 Compute the usual derivative

˜X(t) =ddt

X(γ(t))

its a vector field on Rn but not a tangent vector field on M in general.

2 Project ˜X(t) orthogonally on Tγ(t)M ⊂ R3. The result is the covariantderivative associated with the Levi-Civita Connection (for induced innerproduct)!

curve γ on M can be seen as a curve in Rn. Then1 Compute the usual derivative

˜X(t) =ddt

X(γ(t))

its a vector field on Rn but not a tangent vector field on M in general.2 Project ˜X(t) orthogonally on Tγ(t)M ⊂ R3. The result is the covariant

derivative associated with the Levi-Civita Connection (for induced innerproduct)!

Outline

1 Introduction

Minimizing Properties

• We saw twice the word “geodesics” in different context:

F Riemannian Geodesics: curves of shortest length between two points,

F Geodesics of a Connection, a.k.a Autoparallel curves, i.e. curve with nullcovariant acceleration.

• They are almost the same for The Levi-Civita connection: Levi-Civitageodesics are locally minimizing for the Riemannian induced distance.

• Length functional and Energy functional on a Riemannian manifold M

`(c) =

0|c(t)|c(t) dt E(c) =

0|c(t)|2c(t) dt

• It can be show they have essentially the same minimizer (need somework) – up to parametrization.

• Minimizers of E satisfy the Euler-Lagrange equation

∇c(t)c(t) = 0: The Geodesic Equation!

Solutions have constant velocity norm.

• Rn, trivial connection: c(t) = 0. Trivial solution, straight-line segment

c(t) = P + t~v

• Seems to make sense :-))

`(c) =

0|c(t)|c(t) dt E(c) =

0|c(t)|2c(t) dt

c(t) = P + t~v

`(c) =

0|c(t)|c(t) dt E(c) =

0|c(t)|2c(t) dt

c(t) = P + t~v

`(c) =

0|c(t)|c(t) dt E(c) =

0|c(t)|2c(t) dt

c(t) = P + t~v

`(c) =

0|c(t)|c(t) dt E(c) =

0|c(t)|2c(t) dt

c(t) = P + t~v

Sphere Example

• Arc of great circles through point x, initial velocity v:

t 7→ cos t x + sin t v

• They are the geodesics of the sphere. The green minimizes distancebetween the green points. The red does not.

Christoffel Symbols for the Levi-Civita Connection

• Use the formula

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

for X = ∂xi , Y = ∂xj , Z = ∂xl .

• Develop: ⟨∇∂xi

∂xj , ∂xl

n∑k=1

Γkij 〈∂xk , ∂xl 〉 =

n∑k=1

Γkij gkl

(∂gjl

∂xi+∂gil

∂xj− ∂gij

• Remark [∂xi , ∂xj ] = 0 =⇒ ∇∂xi∂xj = ∇∂xi

∂xj (symmetry) and impliesΓk

ij = Γkji .

• Use the formula

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

for X = ∂xi , Y = ∂xj , Z = ∂xl .

∂xj , ∂xl

n∑k=1

Γkij gkl

(∂gjl

∂xi+∂gil

∂xj− ∂gij

ij = Γkji .

• Use the formula

〈∇X Y ,Z 〉 =12(X 〈Y ,Z 〉 + Y 〈Z ,X 〉 − Z 〈X ,Y 〉

− 〈[Y ,Z ],X 〉 − 〈[X ,Z ],Y 〉 + 〈[X ,Y ],Z 〉)

for X = ∂xi , Y = ∂xj , Z = ∂xl .

∂xj , ∂xl

n∑k=1

Γkij gkl

(∂gjl

∂xi+∂gil

∂xj− ∂gij

ij = Γkji .

• Rewrite asg11 . . . g1n...

...gn1 . . . gnn

︸︷︷︸

Matrix G of the metric

Γ1ij...

∂gj1∂xi

+ ∂gi1∂xj− ∂gij

...∂gjn∂xi

+ ∂gin∂xj− ∂gij

• Solve to get

Γlij =

m∑k=1

g lk(∂gjk

∂xi+∂gik

∂xj− ∂gij

)g lk = (lk)-coefficient of G−1.

• Rewrite asg11 . . . g1n...

...gn1 . . . gnn

︸︷︷︸

Matrix G of the metric

Γ1ij...

∂gj1∂xi

+ ∂gi1∂xj− ∂gij

...∂gjn∂xi

+ ∂gin∂xj− ∂gij

• Solve to get

Γlij =

m∑k=1

g lk(∂gjk

∂xi+∂gik

∂xj− ∂gij

)g lk = (lk)-coefficient of G−1.

Christoffel Symbols for Spherical Coordinates

• Parametrization of S2 by

f : (θ, ϕ) ∈ [0, π)× [0, 2π) 7→ (sin θ cosϕ, sin θ sinϕ, cos θ)

• Jacobian:

cos θ cosϕ − sin θ sinϕcos θ sinϕ sin θ cosϕ− sin θ 0

• Coordinate vector fields:

∂θ =

cos θ cosϕcos θ sinϕ− sin θ

, ∂ϕ =

− sin θ sinϕsin θ cosϕ

• Metric in coordinates:

(〈∂θ, ∂θ〉〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉〈∂ϕ, ∂ϕ〉

(1 00 sin2 θ

• Jacobian:

∂θ =

, ∂ϕ =

(〈∂θ, ∂θ〉〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉〈∂ϕ, ∂ϕ〉

(1 00 sin2 θ

• Jacobian:

∂θ =

, ∂ϕ =

(〈∂θ, ∂θ〉〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉〈∂ϕ, ∂ϕ〉

(1 00 sin2 θ

• Jacobian:

∂θ =

, ∂ϕ =

(〈∂θ, ∂θ〉〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉〈∂ϕ, ∂ϕ〉

(1 00 sin2 θ

• Recall that we need to solve:

Γlij =

m∑k=1

g lk(∂gjk

∂xi+∂gik

∂xj− ∂gij

)• Sympy says :-)(

cot θ

(− sin θ cos θ

)• Geodesic equations:

dt2 + sin θ cos θ(

dt2 + cot θdθdt

Outline

1 Introduction

Exponential Map

• Uniqueness of solutions of the geodesic equation for small t

∇c(t)c(t) = 0, c(0) = P, c(0) = ~v ∈ TPM

• Call solutionΦ(P, ~v , t)

• Easily shown thatΦ(P, α~v , t) = Φ(P, ~v , αt)

Exponential Map

∇c(t)c(t) = 0, c(0) = P, c(0) = ~v ∈ TPM

Exponential Map

∇c(t)c(t) = 0, c(0) = P, c(0) = ~v ∈ TPM

• Instead of small t : small ~v ∈ TPM, t ≤ 1: Mapping ~v 7→ Φ(P, ~v , 1) calledExponential Map

~v 7→ ExpP(~v).

• t 7→ ExpP(t~v) : geodesic with initial position P, initial velocity ~v .

• link with Riemannian distance: d(P,ExpP(~v)) = |~v | (|v | “small”).

~v 7→ ExpP(~v).

Riemannian Log-Map

• ExpP local diffeomorphism between (small) ball centered at 0 ∈ TPMand neighborhood of P in M. Inverse is often called LogP , theRiemannian Log-map.

• How to compute LogP Q?

• Need to find ~v ∈ TPM, such that ExpP(~v) = Q? Two main techniques(might be more).

1 Shooting method: Adapt ~v in function of the error between ExpP(~v) and Q.

2 Path straightening: given a curve joining P and Q, straighten it so itbecomes a geodesic.

Riemannian Log-Map

Shooting

• Guess at step n: ~vn: “shoot” i.e. compute Qn = ExpP ~vn.

• Estimate shooting error between Q and Qn as ~wn ∈ TQn M.• Parallel transport it back to P : P∇( ~wn), correct estimate~vn+1 = ~vn + P∇(~wn).

Shooting

• Estimate shooting error between Q and Qn as ~wn ∈ TQn M.

• Parallel transport it back to P : P∇( ~wn), correct estimate~vn+1 = ~vn + P∇(~wn).

Shooting

• Estimate shooting error between Q and Qn as ~wn ∈ TQn M.• Parallel transport it back to P : P∇( ~wn), correct estimate~vn+1 = ~vn + P∇(~wn).

Path Straightening

• c(t), c(0) = P, c(1) = Q.

• Deform it so as to decrease its covariant acceleration ∇c c.

Path Straightening

• c(t), c(0) = P, c(1) = Q.• Deform it so as to decrease its covariant acceleration ∇c c.

Path Straightening

The Sphere Again!• Geodesic as arc of great circles: (P, ~v) : t 7→ cos

(|~v |t)

P + sin(|~v |t)~v .

• They satisfy the geodesic equation: tangential acceleration is 0!• Exponential map: t = 1:

ExpP(~v) = cos(|~v |)

P + sin(|~v |)~v

• What happens if |~v | = π?, |~v | > π?• Log-map?

• Expected norm of LogPQ: arccos (POQ). Project PQ orthogonally onTPS2: vector ~w . Adapt length so that it becomes arccos (POQ).

(|~v |t)

P + sin(|~v |t)~v .

• They satisfy the geodesic equation: tangential acceleration is 0!

• Exponential map: t = 1:

P + sin(|~v |)~v

(|~v |t)

P + sin(|~v |t)~v .

P + sin(|~v |)~v

(|~v |t)

P + sin(|~v |t)~v .

P + sin(|~v |)~v

• What happens if |~v | = π?, |~v | > π?

• Log-map?

(|~v |t)

P + sin(|~v |t)~v .

P + sin(|~v |)~v

(|~v |t)

P + sin(|~v |t)~v .

P + sin(|~v |)~v

Fréchet Means

• P1, . . . ,Pn point on M. Mean/center of mass?

• In Euclidean space

n∑i=1

• Not possible in Riemannian Manifold: addition of points not defined.

• Alternate definition / characterization:

P = argminQ E(Q) =12

n∑i=1

d(Pi ,Q)2.

with d(P,Q)2 = ‖P −Q‖2.

• For general Riemannian manifold replace ‖ − ‖ by Riemannian distance.

• A minimum if it exists is called a Fréchet Mean of the sample P1, . . . ,Pn.It may fail to exist, it may fail to be unique!

Fréchet Means

• P1, . . . ,Pn point on M. Mean/center of mass?• In Euclidean space

n∑i=1

d(Pi ,Q)2.

with d(P,Q)2 = ‖P −Q‖2.

Fréchet Means

n∑i=1

d(Pi ,Q)2.

with d(P,Q)2 = ‖P −Q‖2.

Fréchet Means

n∑i=1

d(Pi ,Q)2.

with d(P,Q)2 = ‖P −Q‖2.

Fréchet Means

n∑i=1

d(Pi ,Q)2.

with d(P,Q)2 = ‖P −Q‖2.

Fréchet Means

n∑i=1

d(Pi ,Q)2.

with d(P,Q)2 = ‖P −Q‖2.

• Can we compute it?

• f i (Q) = 12 d(Pi ,Q)2 (Karcher 1979?). Gradient?

GradQ f i = − LogQ Pi (?)

(Compare with the Euclidean case!)

• Continuous gradient descent:

dQ(t)dt

= −∇Q(t)E .

• Discrete gradient descent?

Qn+1 = Qn − τ GradQ E?

• No! Adding a point and a tangent vector not defined!

dQ(t)dt

= −∇Q(t)E .

dQ(t)dt

= −∇Q(t)E .

dQ(t)dt

= −∇Q(t)E .

dQ(t)dt

= −∇Q(t)E .

• Exp-map! To get back to the manifold!

• General discrete gradient descent for a function E(Q):

Qn+1 = ExpQn (−τ GradQn E)

Back to Fréchet Means• (?) and Exp-map

Qn+1 = ExpQn

n∑i=1

LogQ Pi

• Choose τ = 1n :

∑ni=1 LogQ Pi : Euclidean average of the vectors LogQ Pis in TQM.

• With that τ , shown to be a Newton-like descent (Pennec 2006). Usuallyvery efficient.

Mean on the sphere. A few lines withPython Numpy or Matlab. Convergesin less than 10 iterations in general.

More complicated manifolds withreasonably efficient Exp and Logmaps (Stefan?)

Qn+1 = ExpQn

n∑i=1

LogQ Pi

)• Choose τ = 1

Qn+1 = ExpQn

n∑i=1

LogQ Pi

)• Choose τ = 1

Outline

1 Introduction

Curvature – Variations of Geodesics – Jacobi Fields

• Curvature tensor: X , Y , Z vector fields:

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

• I’d love to say more about it and links to Jacobi fields, sectionalcurvature, but I am definitely running out of time!

The End

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

The End

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

The End

Bibliography

• Boothby: Introduction to Differential Manifolds and RiemannianGeometry, Wiley.

• do Carmo: Riemannian Geometry, Birkhäuser.• Gallot, Hulin, Lafontaine: Riemannian Geometry, Springer.• Small: The statistical theory of shapes, Springer.• Absil, Mahoni, Sepulchre: Optimization Algorithms On Matrix Manifolds,

Princeton University Press.

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