Elementary differential geometry - PKUbicmr.pku.edu.cn/~dongbin/Conferences/Mini-Course-IG/...Geometry, is cited by most works of the relatively young eld due to its broad coverage

Elementarydifferentialgeometry

ZhengchaoWan

Introduction

Overview

Differentiablemanifolds

Tangentvectors andtangent spaces

Vector fieldsand tensorfields

Connections

Flatness

Riemannianconnection

Submanifolds

Elementary differential geometry

Zhengchao Wan

Peking University

May 9, 2016


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Information geometry

Information geometry is a branch of mathematics that appliesthe techniques of differential geometry to the field ofprobability theory. This is done by taking probabilitydistributions for a statistical model as the points of aRiemannian manifold, forming a statistical manifold. TheFisher information metric provides the Riemannian metric.

Information geometry reached maturity through the work ofShun’ichi Amari and other Japanese mathematicians in the1980s. Amari and Nagaoka’s book, Methods of InformationGeometry, is cited by most works of the relatively young fielddue to its broad coverage of significant developments attainedusing the methods of information geometry up to the year 2000.


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Applications

Information geometry can be applied where parametrizeddistributions play a role. Here an incomplete list:

• statistical inference

• time series and linear systems

• quantum systems

• neural networks

• machine learning

• statistical mechanics

• biology

• statistics

• mathematical finance


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Overview

• Differentiable manifolds

• Tangent vectors and tangent spaces

• Vector fields and tensor fields

• Connections

• Flatness

• Submanifolds

• Riemannian connection


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Manifolds

Definition (Topological manifolds)

A manifold M of dimension m, or m-manifold, is a topologicalspace with the following properties:

• M is Hausdorff,

• M is locally Euclidean of dimension m, and

• M has a countable basis of open sets.

(U, ϕ) is called a coordinate neighborhood of M, where U is anopen set of M and ϕ is a homeomorphism of U to an opensubset of Rm.


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Differentiable manifolds

Definition (C∞-compatible)

Two coordinate neighborhoods (U, ϕ) and (V , ψ) are calledC∞-compatible if U ∩ V nonempty implies that ϕ ◦ ψ−1 andψ ◦ ϕ−1 are diffeomorphisms of the open subsets ϕ(U ∩V ) andψ(U ∩ V ) of Rm.

Definition (Differentiable structure)

A differentiable or C∞ (or smooth) structure on a manifold Mis a family U = (Uα, ϕα) of coordinate neighborhoods suchthat:

• the Uα cover M.

• (Uα, ϕα) and (Uβ, ϕβ) are C∞-compatible for any α, β.

• any coordinate neighborhood (V , ψ) compatible withevery (Uα, ϕα) ∈ U is itself in U .


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Differentiable manifolds

TheoremLet M be a Hausdorff space with a countable basis of opensets. If V = {Vβ, ψβ} is a covering of M by C∞-compatiblecharts, then there is a unique C∞ structure on M containingthese charts.

TheoremFor any two or finite points on a smooth connected manifoldM, there exists a chart (U, ϕ) ∈ U containing them.


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Examples

• Suppose M be an open set of Rm. Let U = M, andϕ : U → Rm be an embedding. (U, ϕ) becomes a C∞

cover on M, which makes up for a differentiable structureon M.

• Suppose f : Rn+1 → R is C∞. If the gradientgradf = ( ∂f

∂x1 , · · · , ∂f∂xn+1 ) never vanishes on a level set

MC = {p ∈ Rn+1; f (p) = c}, then Mc is an n-dimensionalsmooth manifold.

• Suppose M is an m-dimensional manifold with d-structure(Uα, ϕα). Then an open set U of M is also anm-dimensional manifold with d-structure (Vα, ψα), whereVα = U ∩ Uα, ψα = ϕα|Vα .


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Smooth functions and mappings

Definition (Smooth functions)

Let f : M → R be a function on a smooth manifold M, thenwithin a given coordinate neighborhood (U, ϕ), f = f ◦ ϕ−1 isa function from ϕ(U) to R. f is called C∞ at p ∈ U iff f isC∞ at ϕ(p).

Definition (Smooth mappings)

Suppose M and N are two smooth manifolds, and f : M → N isa mapping, p ∈ M. If there exist two coordinate neighborhoods(U, ϕ) and (V , ψ), where p ∈ U, f (p) ∈ V and f (U) ⊂ V . If

f = ψ ◦ f ◦ ϕ−1 : ϕ(U)← ψ(V ) (1)

is a C∞ mapping, then f is called C∞ at p.


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Examples

• Suppose (U, ϕ; x i ) is a chart of M, then U is also amanifold. It’s easy to verify that x i : U → R are smoothfunctions.

• Let I be a closed interval of R, γ : I → M a mapping. Ifthere exists an open interval (a, b) and smooth mappingγ : (a, b)→ M, s.t. I ⊂ (a, b), and γ|I = γ, we call γ asmooth curve on M. I could also be open or half open.


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Tangent vectors

Definition (Tangent vectors)

A tangent vector v at point p ∈ M is a mapping from C∞p to Rthat satisfies:

• ∀f , g ∈ C∞p ,∀λ ∈ R, v(f + λg) = v(f ) + λv(g);

• ∀f , g ∈ C∞p , v(fg) = v(f )g(p) + f (p)v(g).


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Tangent vectors

LemmaSuppose (U; x i ) is a chart containing p in M, and denotex i0 = x i (p). Then ∀f ∈ C∞p , there exist m smooth functionsgi ∈ C∞p , s.t.

gi (p) =∂f

∂x i(p), 1 ≤ i ≤ m, (2)

and for any q near p, we have

f (q) = f (p) +m∑i=1

(x i (q)− x i0)gi (q). (3)


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Tangent vectors

Proof.

f = f ◦ ϕ−1 ∈ C∞ϕ(0)

On some global neighborhood W around x0 = ϕ(p), we have

f (x)− f (x0) =

∫ 1

0

d

dtf (x0 + t(x − x0))dt

=m∑i=1

(x i − x i0)

∫ 1

0

∂ f

∂x i(x0 + t(x − x0))dt.

We let

gi (x) =

∫ 1

0

∂ f

∂x i(x0 + t(x − x0))dt, gi = gi ◦ ϕ.


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Tangent vectors in Rm

Let M = Rm, x0 ∈ Rm. For a vector v ∈ Rm, we defineDv : C∞x0

→ R as: ∀f ∈ C∞x0

Dv f =df (x0 + tv)

dt

∣∣∣t=0

. (4)

Then Dv is a tangent vector of M.Conversely, if any mapping σ : C∞x0

→ R is a tangent vector,then there exists a unique vector v ∈ Rm, such that Dv = σ.


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Tangent vectors in Rm

Proof.Using the lemma before, we have

f (x) = f (x0) +m∑i=1

(x i − x i0)gi (x), (5)

where gi (x0) = ∂f∂x i

(x0). For any constant function λ, we haveσ(λ) = 0. According to Equation (5), we have

σ(f ) =m∑i=1

∂f

∂x i(x0) · σ(x i ).

We let v = (σ(x1), · · · , σ(xm)).


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Natural basis

Suppose p is contained in a coordinate neighborhood (U, ϕ)and ϕ can be written as a coordinate vectorϕ(p) = (x1(p), · · · , xn(p)). Denote ∂

∂x i|p as an operator which

maps f 7→ ( ∂f∂x i

)p , ( ∂ f∂x i◦ϕ)p. It can be verified that ∂

∂x i|p is a

tangent vector.

TheoremDenote TpM as the set of all tangent vectors at p, then TpMforms a vector space which is called the tangent space.Suppose (U, ϕ) is a neighborhood containing p, then

∂

∂x i

∣∣∣p, 1 ≤ i ≤ n, (6)

form a basis of TpM; Moreover, dimTpM = n.


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Natural basis

Proof.

• For any v ∈ TpM and f ∈ C∞p , first we have

f = f (p) +m∑i=1

(x i − x i0)gi , gi (p) =∂f

∂x i(p),

where x i0 = x i (p). Then

v(f ) = v(f (p) +m∑i=1

(x i − x i0)gi )

=m∑i=1

v(x i )∂

∂x i

∣∣∣p(f ).


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Natural basis

Proof.

• For any a1, · · · , am ∈ R, if∑

i ai ∂∂x i|p = 0, then for all

j , 1 ≤ j ≤ m,

0 =( m∑

i=1

ai∂

∂x i

∣∣∣p

)(x j) =

m∑i=1

ai∂x j

∂x i(p) = aj .

Therefore, { ∂∂x i|p : 1 ≤ i ≤ m} is linear independent.


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Differential

Definition (Differential)

Let λ : M → N be a smooth mapping from a manifold M toanother manifold N. Given a tangent vector D ∈ Tp(M) of M,then it can be verified that the mapping D ′ : C∞λ(p) → R

defined by D ′(f ) = D(f ◦ λ) belongs to Tλ(p)N. Representingthis correspondence as D ′ = (dλ)p(D), the linear mapping(dλ)p : TpM → Tλ(p)N is called the differential of λ at p.


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Example

In fact, if we replace the above N with R, then Tλ(p)R = R,and thus (dλ)p ∈ T ∗pM, where T ∗pM is the dual vector space of

TpM. Now, we let the coordinate function x i = λ : M → R.Then

(dx i )p∂

∂x j=( ∂

∂x j

)′

p,

where for any f ∈ C∞x i (p)( ∂

∂x j

)′

p(f ) = f

′(x i (p))

(∂x i∂x j

)∣∣∣p

= f′δij ,

therefore we have 〈dx i , ∂∂x j〉p = dx i ( ∂

∂x j)|p = δij .


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Example

dx i form a basis of T ∗pM. Generally,for any α ∈ T ∗pM, we have

α =m∑i=1

αidxi |p =

m∑i=1

〈 ∂∂x i

∣∣∣p, α〉dx i |p.

Especially, for any f ∈ C∞p ,

df |p =m∑i=1

∂f

∂x i(p)dx i |p.


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Vector fields

Denote TM = ∪p∈M TpM.

A vector field of M is a mapping X : M 7→ TM, such that∀p ∈ M,X (p) ∈ TpM.

For example, in a given coordinate neighborhood (U, ϕ, x i ), ∂∂x i

is a vector field on U.


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Smooth vector fields

Definition (Smooth vector fields)

Let X : M → TM be a vector field on M. X is called a smoothvector field if ∀p ∈ M, there exists a coordinate neighborhood(U, ϕ, x i ) containing p and

X |U =n∑

i=1

X i ∂

∂x i, (7)

such that X i , i = 1, · · · , n are smooth functions on U.

Obviously, the coordinate vector fields ∂∂x i

∣∣∣p

are smooth on U.

Denote T (M) as the set of all smooth vector fields on M.


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Tensors

A (r , s)-tensor τ at p ∈ M is refered to a multilinear map

τ : T ∗pM × · · · × T ∗pM︸︷︷︸r copies

×TpM × · · · × TpM︸︷︷︸s copies

→ R (8)

Define T rs (p) as the set of all (r , s)-tensors at p ∈ M:

T rs (p) = L(T ∗pM, · · · ,T ∗pM︸︷︷︸

r copies

,TpM, · · · ,TpM︸︷︷︸s copies

;R), (9)

thus T rs (p) is a vector space of dimension mr+s .


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Tensors

Under a coordinate neighborhood (U; x i ), T rs (p) has a natural

basis:

∂

∂x i1

∣∣∣p⊗ · · · ⊗ ∂

∂x ir

∣∣∣p⊗ · · · ⊗ dx j1 |p ⊗ · · · ⊗ dx js |p, (10)

1 ≤ i1, · · · , ir , j1, · · · , js ≤ m. (11)

Here if (df )|p ∈ T ∗p (M) and v ∈ Tp(M), then

∂

∂x i

∣∣∣p⊗ dx j |p((df )|p, v) = 〈 ∂

∂x i, df 〉p〈v , dx j〉p. (12)

Now we rewrite T rx (p) as

T rs (p) = TpM ⊗ · · · ⊗ TpM︸︷︷︸

r copies

⊗T ∗p (M)⊗ · · · ⊗ T ∗pM︸︷︷︸s copies

. (13)


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Tensor fields

Denote T rs (M) = ∪p∈M T r

s (p).

A tensor field of M is a mapping τ : M 7→ T rs (M), such that

∀p ∈ M, τ(p) ∈ T rs (p).


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Smooth tensor fields

Definition (Smooth tensor fields)

Let τ : M → T rs M be a (r , s)-tensor field on M. τ is called a

smooth tensor field if ∀p ∈ M, there exists a coordinateneighborhood (U, ϕ, x i ) containing p and

τ |U = τ i1···irj1···js∂

∂x i1⊗ · · · ⊗ ∂

∂x ir⊗ · · · ⊗ dx j1 ⊗ · · · ⊗ dx js , (14)

such that τ i1···irj1···js are smooth functions on U.

Denote T rs (M) as the set of all smooth tensor fields on M.


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Connections

Definition (Connections)

Suppose M is a smooth manifold with dimension m. Aconnection is a map

∇ : T (M)× T (M)→ T (M), (15)

written (X ,Y ) 7→ ∇XY , satisfying the following properties:

1 ∇XY is linear over C∞(M) in X :∇fX1+gX2Y = f∇X1Y + g∇X2Y for f , g ∈ C∞(M);

2 ∇XY is linear over R in Y :∇X (aY1 + bY2) = a∇XY1 + b∇XY2 for a, b ∈ R;

3 ∇ satisfies the following product rule:∇X (fY ) = f∇XY + (Xf )Y for f ∈ C∞(M).


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Example on Rm

Define the Euclidean connection by

∇X (Y j∂j) = (XY j)∂j . (16)

Hence, ∇XY is just the vector field whose components are theordinary directional derivatives of the components of Y in thedirection X .


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Christoffel symbols

Let {Ei} be a local frame for TM on an open subset U ⊂ M.We usually work with a coordinate frame Ei = ∂i . Expand∇Ei

Ej in terms of this same frame:

∇EiEj = Γk

ijEk . (17)

This defines n3 functions Γkij on U, called the Christoffel

symbols of ∇ with respect to this frame.

Let X ,Y ∈ T (U) be expressed in terms of a local frame byX = X iEi ,Y = Y jEj . Then we have

∇XY = (XY k + X iY jΓkij)Ek . (18)


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Vector fields along curves

Let γ : I → M be a curve, where I is an open interval. At anytime t ∈ I , the velocity γ(t) of γ is invariantly defined as(dγ)t(d/dt). It acts on functions by

γ(t)f =d

dt(f ◦ γ)(t). (19)

If we write the coordinate representation of γ asγ(t) = (γ1(t), · · · , γn(t)), then

γ(t) = γ i (t)∂i . (20)

A vector field along a curve γ : I → M is a smooth mapV : I → TM such that V (t) ∈ Tγ(t)M for every t ∈ I . We letT (γ) denote the space of vector fields along γ.


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Extendible vector field

Suppose γ : I → M is a curve, and V ∈ T (M) is a vector fieldon M. For each t ∈ I , let V (t) = Vγ(t). A vector field V alongγ is said to be extendible if it can be extended to a vectorfield V on a neighborhood of the image of γ.

(a) Extendible vector field. (b) Nonextendible vectorfield.


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Covariant derivative along curves

TheoremLet ∇ be a linear connection on M. For each curve γ : I → M,∇ determines a unique operator

Dt : T (γ)→ T (γ) (21)

satisfying the following properties:

1 Linearity over R: Dt(aV + bV ) = aDtV + bDtW fora, b ∈ R.

2 Product rule: Dt(fV ) = f V + fDtV for f ∈ C∞(I ).

3 If V is extendible, then for any extension V of V,

DtV (t) = ∇γ(t)V . (22)

For any V ∈ T (γ), DtV is called the covariant derivative of Valong γ.


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Parallel translationA vector field V along a curve γ is said to be parallel along γwith respect to ∇ if DtV ≡ 0.

Lemma (Parallel Translation)

Given a curve γ : I → M, t0 ∈ I , and a vector V0 ∈ Tγ(t0)M,there exists a unique parallel vector field V along γ such thatV (t0) = V0.

Figure: Parallel translate of V0 along γ.


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Parallel translation

If γ : I → M is a curve and t0, t1 ∈ I , parallel translationdefines an operator

Pt0t1 : Tγ(t0)M → Tγ(t1)M (23)

by setting Pt0t1V0 = V (t1), where V is the parallel translate ofV0 along γ. This is a linear isomorphism between Tγ(t0)M andTγ(t1)M.

LemmaLet ∇ be a linear connection on M, then we have

DtV (t0) = limt→t0

P−1t0t V (t)− V (t0)

t − t0. (24)

In Rm, we have exactly DtV (t0) = limt→t0

V (t)−V (t0)t−t0

.


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Proof.Choose coordinates near γ(t0), and write V (t) = V j(t)∂j neart0. Then by the properties of Dt , since ∂j si extendible

DtV (t0) = V j(t0)∂j + V j(t0)∇γ(t0)∂j

= (V k(t0) + V j(t0)γ i (t0)Γkij(γ(t0)))∂k .

Let W (t0) = P−1t0t1

V (t). Then

W (t0) = W k(t0)∂k = (V k(t) + (t0 − t)V k(t) + o(t0 − t))∂k

Since W (t0) is the parallel translation of V (t),

V k(t) + V j(t)γ i (t)Γkij(γ(t)) = 0,


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Proof.therefore

W (t0) = (V k(t) + (t − t0)V j(t)γ i (t)Γkij(γ(t)) + o(t − t0))∂k

limt→t0

P−1t0t V (t)− V (t0)

t − t0= lim

t→t0

W (t0)− V (t0)

t − t0

= limt→t0

(V k(t)− V k(t0)

t − t0+ V j(t)γ i (t)Γk

ij(γ(t)))∂k

= (V k(t0) + V j(t0)γ i (t0)Γkij(γ(t0)))∂k = DtV (t0).


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Parallel

Definition (Parallel)

Let X ∈ T (M) be a vector field on M. If for any curve γ onM, Xγ : 7→ Xγ(t) is parallel along γ (with respect to ∇), we saythat X is parallel on M (with respect to ∇).

A necessary and sufficient condition for an X = X i∂i to beparallel is that ∇YX = 0 for all Y ∈ T (M), or equivalently that

∂iXk + X jΓk

ij = 0. (25)


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Curvature and torsion

Let ∇ be a connection on M. Then for vector fieldsX ,Y ,Z ∈ T (M), if we define

R(X ,Y )Z , ∇X (∇YZ )−∇Y (∇XZ )−∇[X ,Y ]Z (26)

T (X ,Y ) , ∇XY −∇YX − [X ,Y ], (27)

then these are also vector fields (∈ T (M)). Here, lettingX = X i∂i and Y = Y i∂i , we have defined [X ,Y ] to be thevector field

[X ,Y ] = XY − YX = (X j∂jYi − Y j∂jX

i )∂i (28)

(this does not depend on the choice of coordinate system).


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Curvature and torsion

Now it can be proved that R ∈ T 13 (M) and T ∈ T 1

2 (M) andthey are called curvature tensor field and torsion tensor fieldrespectively. The component expressions of R and T withrespect to coordinate system (U; x i ) are given by

R(∂i , ∂j)∂k = R lijk∂l and T (∂i , ∂j) = T k

ij ∂k , (29)

and these may be computed in the following way:

R lijk = ∂iΓ

ljk − ∂jΓl

ik + ΓlihΓh

jk − ΓljhΓh

ik and (30)

T kij = Γk

ij − Γkji . (31)


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Flat

Definition (Flat)

A connection ∇ of a smooth manifold M is called flat if for anypoint p ∈ M, there exist a coordinate neighborhood (U; x i ),such that ∂i are all parallel on U or equivalently, ∇∂i∂j ≡ 0 or{Γk

ij} vanish on U. Such coordinate system is called an affinecoordinate system.

It can be proved that ∇ is flat iff R = T = 0.

The Euclidean connection ∇ of Rm is flat.It’s easy to see that R l

ijk = −R ljik and T k

ij = −T kji . Hence when

M is 1-dimensional, R = 0 and T = 0 necessarily hold, andtherefore M is flat.


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Affine transformation

Suppose (U, φ; x i ) is another coordinate neighborhood of Mwith Christoffel symbols Γk

ij . When U ∩ U 6= ∅, we have the

following equation on U ∩ U:

Γkij = Γr

pq

∂xp

∂x i∂xq

∂x j∂xk

∂x r+

∂2x r

∂x i∂x j∂xk

∂x r. (32)

If (U, ϕ; x i ) is affine, then we have ∂2x r

∂x i∂x j∂xk

∂x r . Hence (U, φ; x i )

is affine iff ∂2x r

∂x i∂x j= 0. This is equivalent to the condition that

there exist an m×m matrix A and an m−dimensional vector Bsuch that

ϕ(p) = Aφ(p) + B (∀p ∈ U ∩ U), (33)

which is called an affine transformation.


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Riemannian metric

Definition (Riemannian metric)

A Riemannian metric on a smooth manifold M is a 2-tensorfield g ∈ T 0

2 (M) that is symmetric (i.e., g(X ,Y ) = g(Y ,X ))and positive definite (i.e., g(X ,Y ) > 0 if X 6= 0).

A Riemannian metric thus determines an inner product on eachtangent space TpM, which is typically written〈X ,Y 〉 , g(X ,Y ) for X ,Y ∈ TpM.

A manifold M together with a given Riemannian metric g iscalled a Riemannian manifold (M, g).


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Riemannian metric

We define the length or norm of any tangent vector X ∈ TpMto be |X | , 〈X ,X 〉1/2 and the angle between two nonzerovectors X ,Y ∈ TpM to be the unique θ ∈ [0, π] satisfyingcosθ = 〈X ,Y 〉/(|X ||Y |). We say that X and Y areorthogonal if their angle is π/2, or equivalently if 〈X ,Y 〉 = 0.Vectors E1, · · · ,Ek are called orthonormal if thay are of length1 and pairwise orthogonal, or equivalently if 〈Ei ,Ej〉 = δij .

Given a local frame (∂i , · · · , ∂m) for TM, and (dx1, · · · , dxm)is its dual coframe, a Riemannian metric can be written locallyas g = gijdx

i ⊗ dx j , or g = gijdxidx j , if we denote

dx idx j = 12 (dx i ⊗ dx j + dx j ⊗ dx i ) and gij = 〈∂i , ∂j〉.


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One obvious example of a Riemannian manifold is Rn with itsEuclidean metric g , which is just the usual inner product oneach tangent space TxR

n under the natural identificationTxR

n = Rn. In standard coordinates, this can be written inseveral ways;

g =∑i

dx idx i =∑i

(dx i )2 = δijdxidx j . (34)

The matrix of g in these coordinates is thus gij = δij .


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Proposition

Let M,N be two smooth manifolds, and f : M → N a smoothmapping. If ϕ is a smooth (0, r)-tensor field on N, then wehave a smooth (0, r)-tensor field f ∗ϕ on M: forp ∈ M, ∀v1, · · · , vr ∈ TpM,

((f ∗ϕ)(p))(v1, · · · , vr ) = (ϕ(p))((df )p(v1), · · · , (df )p(vr )).

Particularly, if f is an immersion, and h is a Riemannian metricon N, then g = f ∗h is a Riemannian metric on M. g is calledthe induced metric of h.


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Proof.Suppose r = 2. ∀p ∈ M, take its chart (U; x i ) in M and f (p)’schart (V ; yα) in N, s.t. f (U) ⊂ V .

ϕ|V =n∑

α,β=1

ϕα,βdyα ⊗ dyβ, f α = yα ◦ f ,

where ϕα,β = ϕ(

∂∂yα ,

∂∂yβ

). Then we have

(f ∗ϕ)|U =m∑

i ,i=1

(f ∗ϕ)( ∂

∂x i,∂

∂x j

)dx i ⊗ dx j

=∑i ,j

∑α,β

∂f α

∂x i∂f β

∂x j(ϕα,β ◦ f )dx i ⊗ dx j


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Hypersurfaces in Rn+1

Suppose f : N → Rn+1 is an immersion from n-dimensionalmanifold N into Rn+1. We call (f ,N) the immersedhypersurface of Rn. Let h = 〈·, ·〉 be the standard metric onRn+1, and (x1, · · · , xn+1) the Cartesian coordinate, thenh =

∑n+1α=1(dxα)2. Under a chart (U; ui ) on N, let

xα = f α(u1, · · · , un), 1 ≤ α ≤ n + 1. Then we have

g |U =∑α,i ,j

∂f α

∂ui∂f α

∂ujduiduj


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Riemannian connection

Definition (Riemannian connection)

If for all X ,Y ,Z ∈ T (M),

Z 〈X ,Y 〉 = 〈∇ZX ,Y 〉+ 〈X ,∇ZY 〉, (35)

then we say that ∇ is a metric connection with respect to g orRiemannian connection.

If a metric connection is also symmetric, we call it theRiemannian connection.


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Consider a curve γ : t 7→ γ(t) on M and two vector fields Xand Y along γ. We rewrite Equation (35) as

d

dt〈X (t),Y (t)〉 = 〈DtX (t),Y (t)〉+ 〈X (t),DtY (t)〉. (36)

Now if X and Y are both parallel on γ, then the right handside of the equation above is 0.

〈Pγ(X ),Pγ(Y )〉q = 〈X ,Y 〉p. (37)


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Using ∂i , ∂j , ∂k in Equation (35), we have

∂k〈∂i , ∂j〉 = 〈∇∂k∂i , ∂j〉+ 〈∂i ,∇∂k∂j〉,

which is equivalent to

∂kgij = Γhkighj + Γh

kjghi

= Γki ,j + Γkj ,i

where Γki ,j , 〈∇∂k∂i , ∂j〉 = Γhkighj . For Riemannian connection,

which requires Γij ,k = Γji ,k , we have

Γij ,k =1

2(∂igjk + ∂jgki − ∂kgij). (38)


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Flat Riemannian connection

Suppose a Riemannian connection ∇ is flat and there exists anaffine coordinate system [x i ]. Since ∂i is parallel on M, 〈∂i , ∂j〉is constant on M. If

〈∂i , ∂j〉 = δij , (39)

we call such coordiante system a Euclidean coordinatesystem.In fact, the Riemmanian connection is flat iff there exists aEuclidean coordinate system.


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Fundamental lemma

Theorem (Fundamental lemma of Riemannian geometry)

Let (M, g) be a Riemannian manifold. There exists a uniquelinear connection ∇ on M that is compatible with g andsymmetric. (A linear connection ∇ is symmetric if its torsionvanishes identically.)


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Definition (Regular submanifold)

Let F : M → N be a smooth injective map between twosmooth manifolds satisfying the following properties:

1 (dF )p : TpM → TF (p)N is injective for all p ∈ M;

2 F : M → F (M) is a homeomorphism with respect to theinduced topology from N onto F (M).

Then we call F as a regular embedding and M as a regularsubmanifold of N.

We can consider M as a subset of N.


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Proposition

Suppose M is an regular n-dimensional submanifold of anm-dimensional manifold N. For any p ∈ M, there exist slicecoordinates (x1, · · · , xm) on a neighborhood U of p ∈ N s.t.U ∩M is given by {x : xn+1 = · · · = xm = 0}, and(x1, · · · , xn) form local coordinates for M. At each q ∈ U ∩M,TqM can be naturally identified as the subspace of TqNspanned by the vectors (∂1, · · · , ∂n).


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Figure: Slice coordinates.


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Projection of connections

Let N be an n-dimensional manifold and M an m-dimensionalsubmanifold of N. Suppose TpN = TpM ⊕ T⊥p M, where TpM

is the tangent space of p on M, and T⊥p M its orthogonal

complement space with respect to g . Let π> : TN|M → TMdenote the orthogonal projection. Then we define ∇> on M as

(∇>XY )p = π>((∇XY )p). (40)

It can be verified that ∇> is a connection on M, and we callsuch ∇> the projection of ∇ onto M with respect to g .


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Autoparallel submanifolds

Let (U, ϕ; xα) and (V , φ; y i ) be coordinate neighborhoods ofM and N respectively, where F (U) ⊂ V .

Now letting X ,Y ∈ T (M) and X∣∣U

= Xα∂α and Y∣∣U

= Y α∂αbe vector fields on M and ∇ the connection on N, generally,we don’t have ∇XY ∈ T (N) to be a vector field of M.

If, however, ∇XY ∈ T (M) for ∀X ,Y ∈ T (M), that is∇XY = ∇>XY , then we say that M is autoparallel withrespect to ∇ and ∇ can be viewed as the connection on M.


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Using identities such as ∂α = (∂αyi )∂i , we have

∇XY = Xα(∂αYβ)∂β + XαY β∇∂α∂β, (41)

∇∂α∂β = {(∂αy i )(∂βyj)Γk

ij + ∂α∂βyk}∂k . (42)

Thus, M is autoparallel iff ∇∂α∂β ∈ T (M) holds for all α, β.This, in turn, is equivalent to there existing m3 functions{Γγαβ}(∈ C∞(M)) which satisfy

∇∂α∂β = Γγαβ∂γ . (43)

Then we have

Γγαβ∂γyk = (∂αy

i )(∂βyj)Γk

ij + ∂α∂βyk (44)


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Autoparallel flat submanifolds

Suppose N is flat with respect to ∇. Then it can be provedthat autoparallel submanifolds are also flat. Hence without lossof generality we may assume that [xα] and [y i ] are affinecoordinate systems in Equation (44).

Due to flatness, we have ∂α∂βyk = 0, which is equivalent to

there existing an n ×m matrix A and an n-dimensional vectorB that satisfies

φ(p) = Aϕ(p) + B (∀p ∈ M) (45)

A subspace of Rn which may be expressed as{Au + B|u ∈ Rm} is called the affine subspace of Rn.


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Autoparallel flat submanifolds

TheoremIf N is flat, then a necessary and sufficient condition for asubmanifold M to be autoparallel is that M is expressed as anaffine subspace of N with respect to an affine coordinatesystem.


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Open subsets of N are autoparallel.

1-dimensional autoparallel submanifolds are called autoparallelcurves or geodesics. For a curve γ : t 7→ γ(t), the condition inEquation (43) may be rewritten using Equation (22) as

Dt γ(t) = Γ(t)γ(t). (46)

As noted before, connections on 1-dimensional manifolds arenecessarily flat, thus Γ(t) ≡ 0 and Dt γ(t) = 0, which can beexpressed as

γ(t) + γ i (t)γj(t)(Γkij)γ(t) = 0. (47)

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