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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
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
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
Slide 2/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 3/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 3/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 3/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 3/68 — François Lauze — Differential Geometry — September 2014,
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
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?
Slide 4/68 — François Lauze — Differential Geometry — September 2014,
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
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?
Slide 4/68 — François Lauze — Differential Geometry — September 2014,
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
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?
Slide 4/68 — François Lauze — Differential Geometry — September 2014,
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
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?
Slide 4/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 5/68 — François Lauze — Differential Geometry — September 2014,
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
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∑
i=1
X i (p)∂
∂xi
(Xf )p =n∑
i=1
X i (p)∂f∂xi
(p)
• Called Lie Derivative of f along X .
Slide 6/68 — François Lauze — Differential Geometry — September 2014,
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
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∑
i=1
X i (p)∂
∂xi
(Xf )p =n∑
i=1
X i (p)∂f∂xi
(p)
• Called Lie Derivative of f along X .
Slide 6/68 — François Lauze — Differential Geometry — September 2014,
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
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∑
i=1
X i (p)∂
∂xi
(Xf )p =n∑
i=1
X i (p)∂f∂xi
(p)
• Called Lie Derivative of f along X .
Slide 6/68 — François Lauze — Differential Geometry — September 2014,
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
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∑
i=1
X i (p)∂
∂xi
(Xf )p =n∑
i=1
X i (p)∂f∂xi
(p)
• Called Lie Derivative of f along X .
Slide 6/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 7/68 — François Lauze — Differential Geometry — September 2014,
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
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)
Slide 8/68 — François Lauze — Differential Geometry — September 2014,
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
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)
Slide 8/68 — François Lauze — Differential Geometry — September 2014,
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
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)
Slide 8/68 — François Lauze — Differential Geometry — September 2014,
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
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)
t
• In coordinates: X =∑
X i∂xi , Y =∑i
Y ∂xi
[X ,Y ] =∑
i
∑j
X j ∂Y i
∂xj− Y j ∂X i
∂xj
∂xi
Slide 9/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 10/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 10/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 10/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 11/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 12/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 12/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 12/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 13/68 — François Lauze — Differential Geometry — September 2014,
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
Example: Sphere and Stereographic Projection
ΦN : S2\N → R2
(x , y , z) 7→(
x1− z
,y
1− z
) ϕN : R2 → S2
(u, v) 7→ 1`
(2u, 2v , `− 2)
(` = u2 + v2 + 1)
Metric of the sphere via stereographic projection?
Slide 14/68 — François Lauze — Differential Geometry — September 2014,
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
Example: Sphere and Stereographic Projection
ΦN : S2\N → R2
(x , y , z) 7→(
x1− z
,y
1− z
)
ϕN : R2 → S2
(u, v) 7→ 1`
(2u, 2v , `− 2)
(` = u2 + v2 + 1)
Metric of the sphere via stereographic projection?
Slide 14/68 — François Lauze — Differential Geometry — September 2014,
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
Example: Sphere and Stereographic Projection
ΦN : S2\N → R2
(x , y , z) 7→(
x1− z
,y
1− z
) ϕN : R2 → S2
(u, v) 7→ 1`
(2u, 2v , `− 2)
(` = u2 + v2 + 1)
Metric of the sphere via stereographic projection?
Slide 14/68 — François Lauze — Differential Geometry — September 2014,
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
Example: Sphere and Stereographic Projection
ΦN : S2\N → R2
(x , y , z) 7→(
x1− z
,y
1− z
) ϕN : R2 → S2
(u, v) 7→ 1`
(2u, 2v , `− 2)
(` = u2 + v2 + 1)
Metric of the sphere via stereographic projection?
Slide 14/68 — François Lauze — Differential Geometry — September 2014,
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
• Mapping between T(u,v)R2 and TϕN (u,v)S2: JϕN (u,v)
JϕN (u,v) =2`2
`− 2u2 −2uv−2uv `− 2v2
2u 2v
• 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
2v
• ∂u and ∂v tangent vectors to TϕN (u,v)S2 ⇐⇒ ∂u⊥ϕN(u, v), ∂v⊥ϕN(u, v):check it!
Slide 15/68 — François Lauze — Differential Geometry — September 2014,
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
• Mapping between T(u,v)R2 and TϕN (u,v)S2: JϕN (u,v)
JϕN (u,v) =2`2
`− 2u2 −2uv−2uv `− 2v2
2u 2v
• 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
2v
• ∂u and ∂v tangent vectors to TϕN (u,v)S2 ⇐⇒ ∂u⊥ϕN(u, v), ∂v⊥ϕN(u, v):check it!
Slide 15/68 — François Lauze — Differential Geometry — September 2014,
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
• Mapping between T(u,v)R2 and TϕN (u,v)S2: JϕN (u,v)
JϕN (u,v) =2`2
`− 2u2 −2uv−2uv `− 2v2
2u 2v
• 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
2v
• ∂u and ∂v tangent vectors to TϕN (u,v)S2 ⇐⇒ ∂u⊥ϕN(u, v), ∂v⊥ϕN(u, v):check it!
Slide 15/68 — François Lauze — Differential Geometry — September 2014,
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
• 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
)
Slide 16/68 — François Lauze — Differential Geometry — September 2014,
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
• 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
)
Slide 16/68 — François Lauze — Differential Geometry — September 2014,
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
• 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
)
Slide 16/68 — François Lauze — Differential Geometry — September 2014,
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
Riemannian Metric: Charts / Parametrisations
• With local parametrization θ(x) = (x1, . . . , xn)→ M, smooth family ofpositive definite matrices:
gx =
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
Slide 17/68 — François Lauze — Differential Geometry — September 2014,
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
Riemannian Metric: Charts / Parametrisations
• With local parametrization θ(x) = (x1, . . . , xn)→ M, smooth family ofpositive definite matrices:
gx =
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
Slide 17/68 — François Lauze — Differential Geometry — September 2014,
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
Riemannian Metric with Charts
Orthonormal vectors on TPM not transported to standard orthonormal oneson parameter space. Need to locally adapt the metric / inner product.
Slide 18/68 — François Lauze — Differential Geometry — September 2014,
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
With Charts: Fisher Information Metric
• 1D Gaussian distributions: fµ,σ(x) = 1√2πσ2 e−
(x−µ)2
2σ2
• Parametrized by (θ1, θ2) = (µ, σ) ∈ E = R× R++
• Fisher Information Metric on E :(g(µ,σ)
)ij =
∫ ∞−∞
∂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
σ2
Slide 19/68 — François Lauze — Differential Geometry — September 2014,
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
With Charts: Fisher Information Metric
• 1D Gaussian distributions: fµ,σ(x) = 1√2πσ2 e−
(x−µ)2
2σ2
• Parametrized by (θ1, θ2) = (µ, σ) ∈ E = R× R++
• Fisher Information Metric on E :(g(µ,σ)
)ij =
∫ ∞−∞
∂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
σ2
Slide 19/68 — François Lauze — Differential Geometry — September 2014,
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
With Charts: Fisher Information Metric
• 1D Gaussian distributions: fµ,σ(x) = 1√2πσ2 e−
(x−µ)2
2σ2
• Parametrized by (θ1, θ2) = (µ, σ) ∈ E = R× R++
• Fisher Information Metric on E :(g(µ,σ)
)ij =
∫ ∞−∞
∂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
σ2
Slide 19/68 — François Lauze — Differential Geometry — September 2014,
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
With Charts: Fisher Information Metric
• 1D Gaussian distributions: fµ,σ(x) = 1√2πσ2 e−
(x−µ)2
2σ2
• Parametrized by (θ1, θ2) = (µ, σ) ∈ E = R× R++
• Fisher Information Metric on E :(g(µ,σ)
)ij =
∫ ∞−∞
∂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
σ2
Slide 19/68 — François Lauze — Differential Geometry — September 2014,
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
Riemannian Manifold
A differential manifold with a Riemannian metric is a Riemannian manifold.
Slide 20/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 21/68 — François Lauze — Differential Geometry — September 2014,
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
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)?
Slide 22/68 — François Lauze — Differential Geometry — September 2014,
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
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)?
Slide 22/68 — François Lauze — Differential Geometry — September 2014,
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
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)?
Slide 22/68 — François Lauze — Differential Geometry — September 2014,
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
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)?
Slide 22/68 — François Lauze — Differential Geometry — September 2014,
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
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)?
Slide 22/68 — François Lauze — Differential Geometry — September 2014,
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
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)?
Slide 22/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 23/68 — François Lauze — Differential Geometry — September 2014,
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
• 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) =
∫ b
a‖c(t)‖c(t) dt
as in the Euclidean case, by now with variable inner products.
Slide 24/68 — François Lauze — Differential Geometry — September 2014,
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
• 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) =
∫ b
a‖c(t)‖c(t) dt
as in the Euclidean case, by now with variable inner products.
Slide 24/68 — François Lauze — Differential Geometry — September 2014,
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
• 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) =
∫ b
a‖c(t)‖c(t) dt
as in the Euclidean case, by now with variable inner products.
Slide 24/68 — François Lauze — Differential Geometry — September 2014,
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
• 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) =
∫ b
a‖c(t)‖c(t) dt
as in the Euclidean case, by now with variable inner products.
Slide 24/68 — François Lauze — Differential Geometry — September 2014,
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
Example
Blue curve:
c =( 1
2 t), t ∈ [−1, 1].
Red curve d : image of the blue!
`(d) =
∫ 1
−1
√u(t)2 + v(t)2 dt
(u(t)2 + v(t)2 + 1)2 =
∫ 1
−1
dt( 14 + t2 + 1
)2 =89
(sympy...)
Slide 25/68 — François Lauze — Differential Geometry — September 2014,
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
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
∫ 1
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.
Slide 26/68 — François Lauze — Differential Geometry — September 2014,
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
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
∫ 1
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.
Slide 26/68 — François Lauze — Differential Geometry — September 2014,
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
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
∫ 1
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.
Slide 26/68 — François Lauze — Differential Geometry — September 2014,
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
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
∫ 1
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.
Slide 26/68 — François Lauze — Differential Geometry — September 2014,
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
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
∫ 1
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.
Slide 26/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 27/68 — François Lauze — Differential Geometry — September 2014,
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
Acceleration of a Curve
• In Rn
c(0) = limt→0
c(t)− c(0)
t
• 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.
Slide 28/68 — François Lauze — Differential Geometry — September 2014,
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
Acceleration of a Curve
• In Rn
c(0) = limt→0
c(t)− c(0)
t
• 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.
Slide 28/68 — François Lauze — Differential Geometry — September 2014,
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
Acceleration of a Curve
• In Rn
c(0) = limt→0
c(t)− c(0)
t
• 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.
Slide 28/68 — François Lauze — Differential Geometry — September 2014,
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
Acceleration of a Curve
• In Rn
c(0) = limt→0
c(t)− c(0)
t
• 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.
Slide 28/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 29/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 29/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 29/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 29/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 30/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 31/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 32/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 32/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 32/68 — François Lauze — Differential Geometry — September 2014,
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
Christoffel Symbols
• Expansion of ∇ in a coordinate system (x1, . . . , xn):
X =n∑
i=1
X i∂xi , Y =n∑
i=1
Y i∂xi
∇X Y =n∑
i=1
X in∑
j=1
∇∂xiY j∂xj Points 1 and 2
=n∑
i=1
X in∑
j=1
(∂Y j
∂xi∂xj + Y j∇∂xi
∂xj
)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).
Slide 33/68 — François Lauze — Differential Geometry — September 2014,
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
Christoffel Symbols
• Expansion of ∇ in a coordinate system (x1, . . . , xn):
X =n∑
i=1
X i∂xi , Y =n∑
i=1
Y i∂xi
∇X Y =n∑
i=1
X in∑
j=1
∇∂xiY j∂xj Points 1 and 2
=n∑
i=1
X in∑
j=1
(∂Y j
∂xi∂xj + Y j∇∂xi
∂xj
)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).
Slide 33/68 — François Lauze — Differential Geometry — September 2014,
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
Christoffel Symbols
• Expansion of ∇ in a coordinate system (x1, . . . , xn):
X =n∑
i=1
X i∂xi , Y =n∑
i=1
Y i∂xi
∇X Y =n∑
i=1
X in∑
j=1
∇∂xiY j∂xj Points 1 and 2
=n∑
i=1
X in∑
j=1
(∂Y j
∂xi∂xj + Y j∇∂xi
∂xj
)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).
Slide 33/68 — François Lauze — Differential Geometry — September 2014,
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
Christoffel Symbols
• Expansion of ∇ in a coordinate system (x1, . . . , xn):
X =n∑
i=1
X i∂xi , Y =n∑
i=1
Y i∂xi
∇X Y =n∑
i=1
X in∑
j=1
∇∂xiY j∂xj Points 1 and 2
=n∑
i=1
X in∑
j=1
(∂Y j
∂xi∂xj + Y j∇∂xi
∂xj
)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).
Slide 33/68 — François Lauze — Differential Geometry — September 2014,
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
Christoffel Symbols
• Expansion of ∇ in a coordinate system (x1, . . . , xn):
X =n∑
i=1
X i∂xi , Y =n∑
i=1
Y i∂xi
∇X Y =n∑
i=1
X in∑
j=1
∇∂xiY j∂xj Points 1 and 2
=n∑
i=1
X in∑
j=1
(∂Y j
∂xi∂xj + Y j∇∂xi
∂xj
)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).
Slide 33/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 34/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 34/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 34/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivative – Acceleration
• Covariant Derivative: differentiation of vector field X along a curveγ(t) ∈M
Ddt
X (t) = X (t) ∈ Tγ(t)M
• From a connection (with slight abuse...) γ(t) velocity vector field of γ
Ddt
X (t) := (∇γX )γ(t)
• (Covariant) acceleration of γ:
γ(t) :=Ddtγ(t) = (∇γ γ)γ(t)
Slide 35/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivative – Acceleration
• Covariant Derivative: differentiation of vector field X along a curveγ(t) ∈M
Ddt
X (t) = X (t) ∈ Tγ(t)M
• From a connection (with slight abuse...) γ(t) velocity vector field of γ
Ddt
X (t) := (∇γX )γ(t)
• (Covariant) acceleration of γ:
γ(t) :=Ddtγ(t) = (∇γ γ)γ(t)
Slide 35/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivative – Acceleration
• Covariant Derivative: differentiation of vector field X along a curveγ(t) ∈M
Ddt
X (t) = X (t) ∈ Tγ(t)M
• From a connection (with slight abuse...) γ(t) velocity vector field of γ
Ddt
X (t) := (∇γX )γ(t)
• (Covariant) acceleration of γ:
γ(t) :=Ddtγ(t) = (∇γ γ)γ(t)
Slide 35/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Covariant Derivatives and Christoffel Symbols
γ(t) = (x1(t), . . . , xn(t)), γ(t) =n∑i
dx j
dt∂xi v(t) =
n∑i=1
v i (t)∂xi
Ddt
v =n∑
j=1
Ddt
(v j∂xj
)
=n∑
j=1
[dv j
dt∂xj + v j D
dt∂xj
]
=n∑
j=1
[dv j
dt∂xj + v j∇∑n
i=1dxidt ∂xi
∂xj
]
=n∑
j=1
[dv j
dt∂xj +
n∑i=1
v j dx i
dt∇∂xi
∂xj
]
=n∑
k=1
dv k
dt+
n∑i,j=1
v j dx i
dtΓk
ij
∂xk
Slide 36/68 — François Lauze — Differential Geometry — September 2014,
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
Simplest Example – II
• Covariant derivative associated to the (trivial) connection ∇X Y = JY (X )on Rn:
Ddt
f (γ(t)) = Jγ(t)f γ(t) =df (γ(t))
dt
• Christoffel symbols are ≡ 0.
Slide 37/68 — François Lauze — Differential Geometry — September 2014,
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
Simplest Example – II
• Covariant derivative associated to the (trivial) connection ∇X Y = JY (X )on Rn:
Ddt
f (γ(t)) = Jγ(t)f γ(t) =df (γ(t))
dt
• Christoffel symbols are ≡ 0.
Slide 37/68 — François Lauze — Differential Geometry — September 2014,
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
Simplest Example – II
• Covariant derivative associated to the (trivial) connection ∇X Y = JY (X )on Rn:
Ddt
f (γ(t)) = Jγ(t)f γ(t) =df (γ(t))
dt
• Christoffel symbols are ≡ 0.
Slide 37/68 — François Lauze — Differential Geometry — September 2014,
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
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!
Slide 38/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 39/68 — François Lauze — Differential Geometry — September 2014,
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
Parallelism
• X vector field along γ is parallel if its covariant derivative is 0
DXdt
= 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.
Slide 40/68 — François Lauze — Differential Geometry — September 2014,
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
Parallelism
• X vector field along γ is parallel if its covariant derivative is 0
DXdt
= 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.
Slide 40/68 — François Lauze — Differential Geometry — September 2014,
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
Parallelism
• X vector field along γ is parallel if its covariant derivative is 0
DXdt
= 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.
Slide 40/68 — François Lauze — Differential Geometry — September 2014,
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
Parallelism
• X vector field along γ is parallel if its covariant derivative is 0
DXdt
= 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.
Slide 40/68 — François Lauze — Differential Geometry — September 2014,
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
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 :
d2x1
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!
Slide 41/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 42/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 43/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 43/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 43/68 — François Lauze — Differential Geometry — September 2014,
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
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 〉)
Slide 44/68 — François Lauze — Differential Geometry — September 2014,
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
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 〉)
Slide 44/68 — François Lauze — Differential Geometry — September 2014,
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
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 〉)
Slide 44/68 — François Lauze — Differential Geometry — September 2014,
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
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 〉)
Slide 44/68 — François Lauze — Differential Geometry — September 2014,
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
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 〉)
Slide 44/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 45/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 45/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 45/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 45/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 45/68 — François Lauze — Differential Geometry — September 2014,
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
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)!
Slide 46/68 — François Lauze — Differential Geometry — September 2014,
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
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. 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)!
Slide 46/68 — François Lauze — Differential Geometry — September 2014,
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
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. 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)!
Slide 46/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 47/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 48/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 48/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 48/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 48/68 — François Lauze — Differential Geometry — September 2014,
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
• Length functional and Energy functional on a Riemannian manifold M
`(c) =
∫ 1
0|c(t)|c(t) dt E(c) =
∫ 1
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 :-))
Slide 49/68 — François Lauze — Differential Geometry — September 2014,
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
• Length functional and Energy functional on a Riemannian manifold M
`(c) =
∫ 1
0|c(t)|c(t) dt E(c) =
∫ 1
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 :-))
Slide 49/68 — François Lauze — Differential Geometry — September 2014,
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
• Length functional and Energy functional on a Riemannian manifold M
`(c) =
∫ 1
0|c(t)|c(t) dt E(c) =
∫ 1
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 :-))
Slide 49/68 — François Lauze — Differential Geometry — September 2014,
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
• Length functional and Energy functional on a Riemannian manifold M
`(c) =
∫ 1
0|c(t)|c(t) dt E(c) =
∫ 1
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 :-))
Slide 49/68 — François Lauze — Differential Geometry — September 2014,
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
• Length functional and Energy functional on a Riemannian manifold M
`(c) =
∫ 1
0|c(t)|c(t) dt E(c) =
∫ 1
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 :-))
Slide 49/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 50/68 — François Lauze — Differential Geometry — September 2014,
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
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
=12
(∂gjl
∂xi+∂gil
∂xj− ∂gij
∂xl
)
• Remark [∂xi , ∂xj ] = 0 =⇒ ∇∂xi∂xj = ∇∂xi
∂xj (symmetry) and impliesΓk
ij = Γkji .
Slide 51/68 — François Lauze — Differential Geometry — September 2014,
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
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
=12
(∂gjl
∂xi+∂gil
∂xj− ∂gij
∂xl
)
• Remark [∂xi , ∂xj ] = 0 =⇒ ∇∂xi∂xj = ∇∂xi
∂xj (symmetry) and impliesΓk
ij = Γkji .
Slide 51/68 — François Lauze — Differential Geometry — September 2014,
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
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
=12
(∂gjl
∂xi+∂gil
∂xj− ∂gij
∂xl
)
• Remark [∂xi , ∂xj ] = 0 =⇒ ∇∂xi∂xj = ∇∂xi
∂xj (symmetry) and impliesΓk
ij = Γkji .
Slide 51/68 — François Lauze — Differential Geometry — September 2014,
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
• Rewrite asg11 . . . g1n...
...gn1 . . . gnn
︸ ︷︷ ︸
Matrix G of the metric
Γ1ij...
Γnij
=12
∂gj1∂xi
+ ∂gi1∂xj− ∂gij
∂x1
...∂gjn∂xi
+ ∂gin∂xj− ∂gij
∂xn
• Solve to get
Γlij =
12
m∑k=1
g lk(∂gjk
∂xi+∂gik
∂xj− ∂gij
∂xk
)g lk = (lk)-coefficient of G−1.
Slide 52/68 — François Lauze — Differential Geometry — September 2014,
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
• Rewrite asg11 . . . g1n...
...gn1 . . . gnn
︸ ︷︷ ︸
Matrix G of the metric
Γ1ij...
Γnij
=12
∂gj1∂xi
+ ∂gi1∂xj− ∂gij
∂x1
...∂gjn∂xi
+ ∂gin∂xj− ∂gij
∂xn
• Solve to get
Γlij =
12
m∑k=1
g lk(∂gjk
∂xi+∂gik
∂xj− ∂gij
∂xk
)g lk = (lk)-coefficient of G−1.
Slide 52/68 — François Lauze — Differential Geometry — September 2014,
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
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:
Jf =
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
0
• Metric in coordinates:
g =
(〈∂θ, ∂θ〉 〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉 〈∂ϕ, ∂ϕ〉
)=
(1 00 sin2 θ
)
Slide 53/68 — François Lauze — Differential Geometry — September 2014,
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
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:
Jf =
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
0
• Metric in coordinates:
g =
(〈∂θ, ∂θ〉 〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉 〈∂ϕ, ∂ϕ〉
)=
(1 00 sin2 θ
)
Slide 53/68 — François Lauze — Differential Geometry — September 2014,
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
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:
Jf =
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
0
• Metric in coordinates:
g =
(〈∂θ, ∂θ〉 〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉 〈∂ϕ, ∂ϕ〉
)=
(1 00 sin2 θ
)
Slide 53/68 — François Lauze — Differential Geometry — September 2014,
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
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:
Jf =
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
0
• Metric in coordinates:
g =
(〈∂θ, ∂θ〉 〈∂θ, ∂ϕ〉〈∂θ, ∂ϕ〉 〈∂ϕ, ∂ϕ〉
)=
(1 00 sin2 θ
)
Slide 53/68 — François Lauze — Differential Geometry — September 2014,
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
• Recall that we need to solve:
Γlij =
12
m∑k=1
g lk(∂gjk
∂xi+∂gik
∂xj− ∂gij
∂xk
)• Sympy says :-)(
Γ111
Γ211
)=
(00
),
(Γ1
12
Γ212
)=
(0
cot θ
),
(Γ1
22
Γ222
)=
(− sin θ cos θ
0
)• Geodesic equations:
d2θ
dt2 + sin θ cos θ(
dϕdt
)2
= 0
d2ϕ
dt2 + cot θdθdt
dϕdt
= 0
Slide 54/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 55/68 — François Lauze — Differential Geometry — September 2014,
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
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)
Slide 56/68 — François Lauze — Differential Geometry — September 2014,
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
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)
Slide 56/68 — François Lauze — Differential Geometry — September 2014,
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
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)
Slide 56/68 — François Lauze — Differential Geometry — September 2014,
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
• 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”).
Slide 57/68 — François Lauze — Differential Geometry — September 2014,
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
• 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”).
Slide 57/68 — François Lauze — Differential Geometry — September 2014,
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
• 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”).
Slide 57/68 — François Lauze — Differential Geometry — September 2014,
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
• 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”).
Slide 57/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 58/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 58/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 58/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 58/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 58/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 59/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 59/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 59/68 — François Lauze — Differential Geometry — September 2014,
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
Path Straightening
• c(t), c(0) = P, c(1) = Q.
• Deform it so as to decrease its covariant acceleration ∇c c.
Slide 60/68 — François Lauze — Differential Geometry — September 2014,
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
Path Straightening
• c(t), c(0) = P, c(1) = Q.• Deform it so as to decrease its covariant acceleration ∇c c.
Slide 60/68 — François Lauze — Differential Geometry — September 2014,
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
Path Straightening
• c(t), c(0) = P, c(1) = Q.• Deform it so as to decrease its covariant acceleration ∇c c.
Slide 60/68 — François Lauze — Differential Geometry — September 2014,
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
Path Straightening
• c(t), c(0) = P, c(1) = Q.• Deform it so as to decrease its covariant acceleration ∇c c.
Slide 60/68 — François Lauze — Differential Geometry — September 2014,
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
Path Straightening
• c(t), c(0) = P, c(1) = Q.• Deform it so as to decrease its covariant acceleration ∇c c.
Slide 60/68 — François Lauze — Differential Geometry — September 2014,
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
Path Straightening
• c(t), c(0) = P, c(1) = Q.• Deform it so as to decrease its covariant acceleration ∇c c.
Slide 60/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 61/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 61/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 61/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 61/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 61/68 — François Lauze — Differential Geometry — September 2014,
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
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).
Slide 61/68 — François Lauze — Differential Geometry — September 2014,
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
Fréchet Means
• P1, . . . ,Pn point on M. Mean/center of mass?
• In Euclidean space
P =1n
n∑i=1
Pi
• 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!
Slide 62/68 — François Lauze — Differential Geometry — September 2014,
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
Fréchet Means
• P1, . . . ,Pn point on M. Mean/center of mass?• In Euclidean space
P =1n
n∑i=1
Pi
• 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!
Slide 62/68 — François Lauze — Differential Geometry — September 2014,
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
Fréchet Means
• P1, . . . ,Pn point on M. Mean/center of mass?• In Euclidean space
P =1n
n∑i=1
Pi
• 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!
Slide 62/68 — François Lauze — Differential Geometry — September 2014,
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
Fréchet Means
• P1, . . . ,Pn point on M. Mean/center of mass?• In Euclidean space
P =1n
n∑i=1
Pi
• 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!
Slide 62/68 — François Lauze — Differential Geometry — September 2014,
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
Fréchet Means
• P1, . . . ,Pn point on M. Mean/center of mass?• In Euclidean space
P =1n
n∑i=1
Pi
• 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!
Slide 62/68 — François Lauze — Differential Geometry — September 2014,
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
Fréchet Means
• P1, . . . ,Pn point on M. Mean/center of mass?• In Euclidean space
P =1n
n∑i=1
Pi
• 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!
Slide 62/68 — François Lauze — Differential Geometry — September 2014,
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
• 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!
Slide 63/68 — François Lauze — Differential Geometry — September 2014,
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
• 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!
Slide 63/68 — François Lauze — Differential Geometry — September 2014,
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
• 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!
Slide 63/68 — François Lauze — Differential Geometry — September 2014,
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
• 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!
Slide 63/68 — François Lauze — Differential Geometry — September 2014,
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
• 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!
Slide 63/68 — François Lauze — Differential Geometry — September 2014,
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
• Exp-map! To get back to the manifold!
• General discrete gradient descent for a function E(Q):
Qn+1 = ExpQn (−τ GradQn E)
Slide 64/68 — François Lauze — Differential Geometry — September 2014,
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
Back to Fréchet Means• (?) and Exp-map
Qn+1 = ExpQn
(τ
n∑i=1
LogQ Pi
)
• Choose τ = 1n :
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?)
Slide 65/68 — François Lauze — Differential Geometry — September 2014,
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
Back to Fréchet Means• (?) and Exp-map
Qn+1 = ExpQn
(τ
n∑i=1
LogQ Pi
)• Choose τ = 1
n :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?)
Slide 65/68 — François Lauze — Differential Geometry — September 2014,
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
Back to Fréchet Means• (?) and Exp-map
Qn+1 = ExpQn
(τ
n∑i=1
LogQ Pi
)• Choose τ = 1
n :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?)
Slide 65/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 66/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 67/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 67/68 — François Lauze — Differential Geometry — September 2014,
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
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
Slide 67/68 — François Lauze — Differential Geometry — September 2014,
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
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.
Slide 68/68 — François Lauze — Differential Geometry — September 2014,