Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection Riemannian manifolds Ravi N Banavar [email protected]1 1 Systems and Control Engineering, IIT Bombay, India March 17, 2017 Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
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Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
1Systems and Control Engineering,IIT Bombay, India
March 17, 2017
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Outline
1 Differentiating a vector field along a curve
2 The covariant derivative
3 Affine connection
4 Riemannian connection
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Outline
1 Differentiating a vector field along a curve
2 The covariant derivative
3 Affine connection
4 Riemannian connection
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Differentiating a vector field along a curve
A curve in R3
• Let α(t) be a curve in R3 parametrized with time (t ∈ R.) LetFi(t); i = 1, 2, 3 be a basis parametrized with time .
• V is a vector field along α. So V (α(t)) ∈ Tα(t)R3.
• Let
V (t) =
3∑i=1
bi(t)Fi(t)
• Then
dV
dt=
3∑i=1
[dbi(t)
dtFi(t) + bi(t)
dFi(t)
dt]
• If the basis is constant, say Fi(t) = ∂∂xi, then
dV
dt=
3∑i=1
[dbi(t)
dt
∂
∂xi]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Frenet-Serre frames
Parametrizing with arclength
• Arclength s (an alternate parametrization)
s4=
∫ t
t0
⟨˙α(t), ˙α(t)
⟩0.5
dt
• The following equalities follow
ds
dt=⟨
˙α(t), ˙α(t)⟩0.5
dα(s(t))
dt=dα
ds
ds
dt⇒ dα
ds= [
dα
dt]/[ds
dt]
• Define
T (s)4=dα
dsunit tangent vector
•
‖T (s)‖ = 1⇒ d
ds〈T (s), T (s)〉 =
⟨2dT
ds, T
⟩= 0
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Curvature and torsion
• Define k(s) =∥∥ dTds
∥∥ - the curvature.
• Define an orthonormal basis as
T (s), N(s), B(s)
where dTds
= k(s)N(s) and B(s) is a third, unit orthonormal vector thatpreserves orientation. (k(s) 6= 0.)
• Let F1 = T, F2 = N,F3 = B (the moving basis). Then
dT
ds= k(s)N(s)
dN
ds= −k(s)T + τ(s)B
dB
ds= −τ(s)N
• τ(s) denotes the torsion (a measure of the out-of-plane bending of thecurve.)
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Outline
1 Differentiating a vector field along a curve
2 The covariant derivative
3 Affine connection
4 Riemannian connection
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
A metric on the tangent space
Notion of distanceThe need to define a notion of distance or the length of a curve on amanifold.
A metricA Riemannian metric on a differentiable manifold is a smoothly varyinginner-product on the tangent space G(x)(·, ·) (also termed a covarianttwo-tensor), that satisfies the following properties. For each x ∈M ,G(x)(·, ·) : TxM × TxM→R satisfies
• G(x)(v, v) ≥ 0 ∀v ∈ TxM and G(x)(v, v) = 0 iff v = 0. Positivedefinite
• G(x)(v, w) = gx(w, v) ∀v, w ∈ TxM Symmetric
• G(x)(α1v1 + α2v2, w) = α1G(x)(v1, w) + α2G(x)(v2, w) ∀v1, v2, w ∈TxM, ∀α1, α2 ∈ R Linearity (in fact bilinear)
A smooth manifold endowed with a Riemannian metric is a Riemannianmanifold.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Length of a curve
Let M be a Riemannian manifold and α(·) be a smooth curve from[a, b] ⊂ I→M . Then the length of the curve is defined as
L(α)4=
∫ b
a
√G(α(t))(
dα
dt,dα
dt)dt
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Rn
The Riemannian metric on Rn is the familiar Euclidean metric.
G(x)(v, w) = vT Iw where I is the n× n identity matrix.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Sphere S2Coordinates (θ, φ) ∈ ((0, π), [0, 2π]).
Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, φ) = (sin θ cosφ, sin θ sinφ, cos θ)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3
vθ∂
∂θ+ vφ
∂
∂φ→[Df(θ, φ)]
[vθvφ
]= [(cos θ cosφ)vθ − (sin θ sinφ)vφ]
∂
∂x+ [(cos θ sinφ)vθ + (sin θ cosφ)vφ]
∂
∂y
−(sin θ)vθ∂
∂z
G(θ, φ)(v, w) =[vθ vφ
] [ 1 00 sin2 θ
] [wθwφ
]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Sphere S2Coordinates (θ, φ) ∈ ((0, π), [0, 2π]).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, φ) = (sin θ cosφ, sin θ sinφ, cos θ)
Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3
vθ∂
∂θ+ vφ
∂
∂φ→[Df(θ, φ)]
[vθvφ
]= [(cos θ cosφ)vθ − (sin θ sinφ)vφ]
∂
∂x+ [(cos θ sinφ)vθ + (sin θ cosφ)vφ]
∂
∂y
−(sin θ)vθ∂
∂z
G(θ, φ)(v, w) =[vθ vφ
] [ 1 00 sin2 θ
] [wθwφ
]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Sphere S2Coordinates (θ, φ) ∈ ((0, π), [0, 2π]).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, φ) = (sin θ cosφ, sin θ sinφ, cos θ)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3
vθ∂
∂θ+ vφ
∂
∂φ→[Df(θ, φ)]
[vθvφ
]= [(cos θ cosφ)vθ − (sin θ sinφ)vφ]
∂
∂x+ [(cos θ sinφ)vθ + (sin θ cosφ)vφ]
∂
∂y
−(sin θ)vθ∂
∂z
G(θ, φ)(v, w) =[vθ vφ
] [ 1 00 sin2 θ
] [wθwφ
]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Cylinder S1 ×R1
Coordinates (θ, z) ∈ ([0, 2π), (−∞,∞)).
Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, z) = (cos θ, sin θ, z)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3
vθ∂
∂θ+ vz
∂
∂z→[Df(θ, φ)]
[vθvφ
]= [− sin θvθ]
∂
∂x+ [cos θvθ]
∂
∂y+ vz
∂
∂z
G(θ, φ)(v, w) =[vθ vz
] [ 1 00 1
] [wθwz
]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Cylinder S1 ×R1
Coordinates (θ, z) ∈ ([0, 2π), (−∞,∞)).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, z) = (cos θ, sin θ, z)
Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3
vθ∂
∂θ+ vz
∂
∂z→[Df(θ, φ)]
[vθvφ
]= [− sin θvθ]
∂
∂x+ [cos θvθ]
∂
∂y+ vz
∂
∂z
G(θ, φ)(v, w) =[vθ vz
] [ 1 00 1
] [wθwz
]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
Cylinder S1 ×R1
Coordinates (θ, z) ∈ ([0, 2π), (−∞,∞)).Expressing the coordinates in R3, in terms of (x, y, z), we have(x, y, z) = f(θ, z) = (cos θ, sin θ, z)Choose the metric induced by the Euclidean metric as follows: Firsttransform a tangent vector to the coordinates in R3
vθ∂
∂θ+ vz
∂
∂z→[Df(θ, φ)]
[vθvφ
]= [− sin θvθ]
∂
∂x+ [cos θvθ]
∂
∂y+ vz
∂
∂z
G(θ, φ)(v, w) =[vθ vz
] [ 1 00 1
] [wθwz
]
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Examples of Riemannian metrics
The Lie group SO(3)
The Riemannian metric on SO(3) is induced by the inner product on theLie algebra so(3), wherein
G(R)(vR, wR)4=⟨
Ω1, Ω2
⟩so(3)
= −1
2Trace(Ω1Ω2) (1)
where vR = RΩ1 ∈ TR SO(3) and wR = RΩ2 ∈ TR SO(3), the lefttranslations by R of Ω1 and Ω2, respectively.Here Ω1 and Ω2 belong to theLie algebra so(3). Note that we have used the left-invariant property of thevector field on SO(3).
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Simple mechanical systems and Riemannian manifolds
For a simple mechanical system, the kinetic energy is a Riemannian metric .
Minimizing the KE for a mechanical system between any two fixed intialand final points is the equivalent to finding a shortest path or a geodesicbetween two points on a Riemannian manifold with the metric being thekinetic energy.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Riemannian manifolds as submanifolds of Rn
• Suppose M is a Riemannian manifold of dimension m immersed in Rn(n ≥ m.)
• Let α(·) be a smooth curve from [a, b](⊂ R)→M . Let V (α(t)) be asmooth vector field defined on the curve α(·). This means, for every t
V (t) ∈ Tα(t)M
• Question What is dVdt
?
• Though V (t) ∈ Tα(t)M , dVdt
need not be in Tα(t)M .
• If we now look at the point α(t) in Rn, the tangent space to Rn at anypoint is Rn and is hence n-dimensional.
• However, we are concerned about the rate of change of V (t) along themanifold. Hence we define a new quantity called the covariantderivative
DV
dt
4= π(
dV
dt) where π denotes the projection on Tα(t)M .
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Properties of the covariant derivative
The covariant derivative satisfies
•D
dt(V +W ) =
DV
dt+DW
dt•
D(fV )
dt=df
dtV + f
DV
dt•
d(g(V,W ))
dt= g(
DV
dt,W ) + g(V,
DW
dt)
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Formulae for the covariant derivative
• Let u4= (u1, . . . , um) be local coordinates for M around a point φ(p),
where φ−1 is a parametrization
φ−1 : Rm→Rn x = φ−1(u) = (g1(u), . . . , gn(u))
• The associated tangent map is
φ−1∗ : TM→TRn
• The basis ∂∂u1 , . . . ,
∂∂um for the tangent space to M at φ(p) in the
coordinates u, can be expressed in terms of the basis ∂∂x1
, . . . , ∂∂xn
using the tangent map as
Fjp4= (φ−1
∗ )φ(p)∂
∂uj=
n∑k=1
[∂gk
∂uj]φ(p)
∂
∂xk
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Computing DFkdt
• Let α be a curve and V be a vector field defined along α.
• Using the basis F1(t), · · · , Fm(t), we express V (t) =∑mk=1 b
k(t)Fk(t),where bk(t) is a smooth function.
• Now
DV (t)
dt=
m∑k=1
dbk
dtFk(t) + bk(t)
DFkdt
• Our interest is to compute DFkdt
.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Computing DFidt
• Recall
Fjp4= (φ−1
∗ )φ(p)∂
∂uj=
n∑k=1
[∂gk
∂uj]φ(p)
∂
∂xk
• ThendFidt
=
n∑k=1
m∑l=1
[∂2gk
∂ul∂ui]φ(p)
dul
dt
∂
∂xk
• Taking the projection π,
DFidt
= πp(dFidt
) =n∑k=1
m∑l=1
[∂2gk
∂ul∂ui]φ(p)
dul
dtπp[
∂
∂xk]
• Express
πp[∂
∂xk] =
m∑j=1
ajkFj
• Define
Γjli4=
n∑k=1
[∂2gk
∂ul∂ui]φ(p)a
jk
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
The Christoffel symbols
• Then
DFidt
=m∑j=1
m∑l=1
Γjlidul
dtFj
• The covariant derivative
DV (t)
dt=
m∑k=1
dbk
dtFk(t) + bk(t)
m∑j=1
m∑l=1
Γjlkdul
dtFj
=m∑k=1
[dbk
dt+
m∑j=1
m∑l=1
Γkljdul
dtbj(t)]Fk
• The Christoffel symbol Γklj denotes the kth component of the covariantderivative of Fl along the curve in which only the jth coordinate isallowed to vary.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Computing Christoffel symbols
• Compute the Christoffel symbols from the Riemannian metric as
Γkij =1
2Gkl(∂Gil
∂xj+∂Gjl∂xi
− ∂Gij∂xl
)
where Gkl stands for the inverse of .
•
∇ ∂∂xi
∂
∂xj= Γkij
∂
∂xk
Γkij are the n3 Christoffel symbols for ∇ in the specified coordinates.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
A vector field along another vector field
Covariant derivative of Y with respect to X
∇XY = (∂Y k
∂xiXi + ΓkijX
iY j)∂
∂xk
Covariant derivative of Y along a curve γ
∇γ(t)Y (t) = (Y k(t) + Γkij(γ(t))xi(t)Y j(t))∂
∂xk
t→(x1(t), . . . , xn(t)) is a local representation of γ.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Outline
1 Differentiating a vector field along a curve
2 The covariant derivative
3 Affine connection
4 Riemannian connection
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
An affine connection
General Riemannian manifoldsHow do we introduce the notion of a covariant derivative in a more generalRiemannian manifold (not necessarily submersed in Rn ?) We need thenotion of an affine connection.
The notion of a connection provides a tool for differentiating vectors alongcurves; in particular, we can talk of the acceleration of a curve in M .
DefinitionAn affine connection ∇ on a differentiable manifold is a mapping
∇ : X (M)×X (M)→X (M)
denoted by (X,Y )→∇XY , which satisfies the following properties
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Covariant derivative
The connection helps us to define the derivative of a vector field alongcurves called the covariant derivative.
DefinitionLet M be a differentiable manifold with an affine connection ∇. Thereexists a unique corespondence which associates to a vector field V along adifferentiable curve α : I→M , another vector field DV
dtalong α, called the
covariant derivative of V along α, such that
• Ddt
(V +W ) = DVdt
DWdt
• D(fV )dt
= dfdtV + f DV
dt
• If V is induced by a vector field Y ∈ X (M), say V (t) = Y (α(t)), then
DV
dt= ∇ dα
dtY
Unlike the Lie derivative, the covariant derivative brings in additionalstructure to the manifold and is not instrinsic to the manifold itself.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Parallelism
DefinitionLet M be a differentiable manifold with an affine connection ∇. A vectorfield V along a curve α : I→M is called parallel when DV
dt= 0, for all t ∈ I.
GeodesicsLet M be a differentiable manifold with an affine connection ∇. A curveα : I→M is called a geodesic if ∇ dα
dt
dαdt
= 0 for all t ∈ I.
Motion in R3
Look at the motion of a point mass in R3. Call its trajectory α(t). Then itsvelocity vector is dα
dt. For what type of α is
D
dt(dα
dt) = 0 ?
Straight lines !. α(t) = k1t+ k2.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Outline
1 Differentiating a vector field along a curve
2 The covariant derivative
3 Affine connection
4 Riemannian connection
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Metric compatible connection
DefinitionA connection on a Riemannian manifold is said to be compatible with themetric g(·, ·), when for any smooth curve α and any pair of parallel vectorfields V and W (i. e. DV
dt= 0, DW
dt= 0) along α, we have
gα(t)(V (α(t)),W (α(t))) = constant
TheoremA connection ∇ on a Riemannian manifold is said to be compatible with themetric g(·, ·) if and only if
X(g(Y,Z)) = g(∇XY,Z) + g(Y,∇XZ)
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN
Differentiating a VF along a curve Covariant derivative Affine connection Riemannian connection
Riemannian connection
DefinitionA connection ∇ on a smooth manifold is said to be symmetric when
∇XY −∇YX = [X,Y ]
Riemannian connection
TheoremThere exists a unique affine connection ∇ on a Riemannian manifold Mthat is both symmetric and is compatible with the Riemannian metric. Thisis called the Levi-Civita (or Riemannian) connection.
Lectures on Riemannian geometry (IIT-B/IIT-GN) March 2017, IIT-GN