MA4C0 Di erential Geometry - University of Warwick · PDF fileMA4C0 Di erential Geometry...

MA4C0 Differential Geometry

April 9, 2013

Contents

1 Introduction 2

2 Prerequisite Manifold Theory 3

3 Prerequisite linear algebra 4

4 New Bundles from Old 6

5 Tensors 7

6 Connections 9

7 Riemannian Metrics 11

8 Levi-Civita connection 13

9 Geodesics, lengths of paths, Riemannian Distance 17

10 Curvature 26

11 Conquering Calculation 30

12 Second Variation Formula 31

13 Classical Riemannian Geometry Results: Bonnet-Myers 33

These notes are based on the 2012 MA4C0 Differential Geometry course, taught byPeter Topping, typeset by Matthew Egginton.No guarantee is given that they are accurate or applicable, but hopefully they will assistyour study.Please report any errors, factual or typographical, to [email protected]

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MA4C0 Differential Geometry Lecture Notes Autumn 2012

1 Introduction

The following is a very sketchy background in certain parts of the course, included to givesome context to what is done. None of the definitions and theorems are stated rigorously.We briefly look at the following

1. manifolds with a notion of length/angle

2. way of differentiating vector fields

3. shortest path between two points

4. curvature

Definition 1.1 (Manifold) A manifold is a topological space such that each point has aneighbourhood that is homeomorphic to a ball in Rn.

Some examples of manifolds are R, S1, S2, T 2, RPn, SO(n). An example that isn’t amanifold is where three half lines meet at the same point. The point is where it fails, as atthis point it is not homeomorphic to a line.

1.1 Intrinsic Curvature of a Surface (Gauss Curvature)

1.1.1 Geodesic Flow

Curvature is very closely connected with the stability of geodesic flow. A positive curvatureencourages geodesics to focus, whereas a negative curvature encourages geodesics to diverge.This is seen on the sphere, where the geodesics from a point focus on the antipodal point.

1.1.2 Volume Growth

Consider the function F : r → V ol(B(p, r)). By comparing this function with the samefunction in Rn we can measure curvature, for example if the manifold is positively curvedthen the function F is less than the equivalent Euclidean function.

1.1.3 Topology

Theorem 1.2 (Gauss Bonnet) If M is a compact surface and K is the Gauss curvaturethen ∫

MKdV = 4π(1− g)

Observe that there is no geometry involved in the right hand side, there is only topology

Theorem 1.3 (Bit of Uniformisation theorem) Every compact surface M has a Rie-mannian metric with constant curvature

• +1 if M is the sphere

• 0 if g = 1

• -1 if g ≥ 2

Theorem 1.4 (Cartan -Hadamard) SupposeM is a Riemannian manifold which is sim-ply connected with curvature less than 0. Then M is diffeomorphic to Rn.

Theorem 1.5 (Bonnet’s) Suppose M is a complete Riemannian manifold , with sectionalcurvature at least λ > 0. Then M is compact.

Note that one could define the Gauss curvature K on a surface at p by length(∂Br) =2πr(1− K

6 r2 +O(r2))

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2 Prerequisite Manifold Theory

Definition 2.1 A smooth manifold M of dimension n is a Hausdorff topological spacewhich is second countable, together with an open cover Uαα∈I ofM and maps φα : Uα → Rnhomeomorphic onto their image such that whenever Uα∩Uβ 6= ∅ then φαφ−1

β : φβ(Uα∩Uβ)→Rn is smooth

The φα or (Uα, φα) are called charts, and φα φ−1β are called transition maps. The

collection of charts is called an atlas.

Definition 2.2 1. A function f :M→ R is smooth if for any chart (Uα, φα) we havef φ−1

α : φα(Uα)→ R is smooth as a function R→ R.

2. A map F : M → M between manifolds is smooth if for any charts φβ F φ−1α is

smooth.

3. A map F : M → M is a diffeomorphism if it is smooth and also a bijection withsmooth inverse.

4. A map F : [−1, 1] → M is smooth if its restriction to (−1, 1) is smooth and all(partial) derivatives are continuous up to the boundary.

Definition 2.3 A tangent vector X at a point p ∈M is a linear functional on the space ofsmooth functions in a neighbourhood of p which can be written locally as f →

∑ni=1 ai

∂f∂xi

(p)

Note that the ai in the above definition depend on the chart used to define the coordinates.

Definition 2.4 The pushforward u?(X) of a tangent vector X at p ∈ M under a mapu :M→N is the vector at u(p) defined, for all smooth f , by

(u?(X))f = X(f u)

for f ∈ C∞(N ).

We can see this operation as a linear map from vectors at p to vectors at u(p). Thetangent space TpM is the space of all vectors at p. This linear map is normally written asdu : TpM → Tu(p)N .

Definition 2.5 u : M → N is an immersion if for all p ∈ M we have that du : TpM →Tu(p)N is injective. It is a submersion if du is surjective for all p and it is an embeddingif it is an immersion and a topological embedding.

2.1 Tangent Bundle and Vector Bundles

The tangent bundle is the space of all points p ∈ U and tangent vectors X at p, i.e. it is allpairs (p,X). This will be a 2n dimensional manifold where n = dimM, and will have someextra structure; that of a vector bundle.

Definition 2.6 A tangent vector field is an assignment to each point p ∈M of a vectorin TpM, which can be written ai(x) ∂

∂xifor smooth local coefficients ai(x).

Definition 2.7 A smooth real vector bundle (E,M, π) of rank k ∈ N0 is a smooth man-ifold E of dimension m+ k, a smooth manifold M of dimension m and a smooth surjectivemap π : E →M such that

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1. there is an open cover Uα of M and diffeomorphisms ψα : π−1(Uα)→ Uα × Rk

2. for all p ∈M, ψα(π−1(p)) = p × Rk

3. whenever Uα ∩Uβ 6= ∅, the map ψα ψ−1β : (Uα ∩Uβ)×Rk → (Uα ∩Uβ)×Rk takes the

form (x, v) 7→ (x,Aαβv) where Aαβ : Uα ∩ Uβ → GL(k,R) and is smooth.

Remark If the rank is zero then E =M up to diffeomorphism.

Definition 2.8 Ep := π−1(p) is called the fibre over p, and has a natural vector spacestructure.

We often write Eπ→M instead of the triple.

Definition 2.9 A smooth bundle morphism (map) H from (E,M, π) to (E′,M′, π′) isa smooth map H : E → E′ which maps Ep to some E′p′ as a linear map

Definition 2.10 Two bundles (E,M, π) and (E′,M ′, π′) are isomorphic if there is asmooth bundle map H : (E,M, π) → (E′,M′, π′) which has an inverse that is a smoothbundle map

Remark We only consider vector bundles up to isomorphism.The tangent bundle is a vector bundle of rank m = dimM. Given a manifold M with

charts (Uα, φα) then the total space TM is the space of pairs (p,X) with p ∈ M and X ∈TpM and π(p,X) = p and the local trivialisations ψα are defined by ψα : π−1(Uα)→ Uα×Rmgiven by (p, ai(x) ∂

∂xi) 7→ (p, a1, ..., am)

Definition 2.11 A smooth section of a vector bundle (E,M, π) is a smooth map s :M→E such that π s = IdM. The space of all sections is denoted Γ(M).

In other words the smooth sections are assignments of a vector in E to each point p ∈M.

3 Prerequisite linear algebra

It is presumed that one knows dual spaces.

3.1 Upper and Lower indices

Let V be a vector space and V ? be the dual space. We have a convention on indices. Wewrite bases of V as ei, with a lower index. We write a general element as aiei, with anupper index on the coefficients. We write bases of V ? as θi, with upper indices, and ageneral element as biθ

i. To understand why, see question 1.10 in problems.

3.2 Inner products and dual spaces

We can equip V with an inner product 〈, 〉. This then induces an isomorphism Θ : V → V ?

by [Θ(v)](w) = 〈v, w〉 for v, w ∈ V . By Θ(v) we mean an element in the dual space, nothingmore. This allows us to extend the inner product to V ? by

〈Θ(v),Θ(w)〉 := 〈v, w〉

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3.3 Tensor Products

Given two vector spaces V and W how can we combine them? One possibility is V ⊕W ,and another is V ⊗W . We consider the latter.

Define F (V,W ) to be the free vector space over R whose generators are elements ofV ×W . Let R(V,W ) be the subspace generated by elements of the form

1. (λv,w)− λ(v, w)

2. (v, λw)− λ(v, w)

3. (v1 + v2, w)− (v1, w)− (v2, w)

4. (v, w1 + w2)− (v, w1)− (v, w2)

Now define V ⊗W = F (V,W )/R(V,W ) and write the coset containing (v, w) by v ⊗ w. Inpractise this means that V ⊗W can be considered as all finite linear combinations of v ⊗ wwhere we agree that (λv)⊗w = λv⊗w = v⊗ (λw) etc following the above. If eii is a basisfor V and fjj∈J is a basis for W then ei⊗ejI,J is a basis for V ⊗W . However, one shouldnote that not every element of V ⊗W can be written v ⊗ w, for example v1 ⊗ w1 + v2 ⊗ w2

for v1, v2 and w1, w2 not linearly dependent in their respective spaces.We will often consider tensors of the type (p, q) ∈ N0×N0, i.e. elements of V ⊗ ...⊗ V ⊗

V ? ⊗ ...⊗ V ? with p V s and q V ?s. We will also focus on the case V = TpM. Such a tensorcan be written as

Ti1,...,ipj1,...,jq

ei1 ⊗ ...⊗ eip ⊗ θj1 ⊗ ...⊗ θjqand for brevity we often only write the coefficients T .

3.4 Extending inner product to tensors

Given 〈, 〉 on V we can define an inner product on V ⊗ ...⊗ V ⊗ V ? ⊗ ...⊗ V ? by

〈v1⊗ ...⊗ vp⊗L1⊗ ...⊗Lq, v1⊗ ...⊗ vp⊗ L1⊗ ...⊗ Lq〉 = 〈v1, v1〉...〈vp, vp〉〈L1, L1〉...〈Lq, Lq〉

i.e. this is the natural way to do it.

3.5 Contraction and Traces

Given an element V ? ⊗ V , a sum of terms like L⊗ v we can take a contraction by takingeach term L ⊗ v and replace with L(v), for example T ji θ

j ⊗ ei then the contraction is T ii .Given an element of V ⊗ V (or V ⊗ V ) we can use the isomorphism Θ : V → V ? to turn itinto an element of V ? ⊗ V and then contract. This is called taking a trace. Given a (p, q)tensor you can contract/trace 2 entries to give either (p− 2, q)or (p− 1, q − 1) or (p, q − 2)tensor.

One can only trace when there is an isomorphism V → V ?. This is apparent later.

3.6 Symmetric and Anti symmetric tensors

Definition 3.1 We define We call a tensor in V ⊗ V symmetric if it is invariant if weswitch the entries of each term. Also T = T ijei ⊗ ej is symmetric if T ij is a symmetricmatrix. More generally a (p, 0) tensor is symmetric if it is invariant when we switch anytwo entries. The vector space of all such tensors is written Symp(V ).

An example of this is v1 ⊗ v2 + v2 ⊗ v1.

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Definition 3.2 A (p, 0) tensor is antisymmetric or alternating or skew symmetric ifswitching any two entries reverses the sign. The vector space of all such tensors is denotedΛpV

An example of this is v1 ⊗ v2 − v2 ⊗ v1.

3.7 Natural Projections

The natural projection from V ⊗ V → Sym2(V ), written sym, is defined by v1 ⊗ v2 7→12(v1 ⊗ v2 + v2 ⊗ v1) and extends naturally to (p, 0) tensors.

The natural projection from V ⊗ V → Λ2(V ) , written Alt is defined by v1 ⊗ v2 7→12(v1 ⊗ v2 − v2 ⊗ v1) and again extends naturally to (p, 0) tensors.

3.8 Exterior or Wedge product

Here we combine ω ∈ Λk(V ) and η ∈ Λl(V ) to give a tensor in Λk+l(V ) by

ω ∧ η := Alt(ω ⊗ η)(k + l)!

k!l!

where the k!l! is just a normalising number that the lecturer has chosen.

4 New Bundles from Old

4.1 Dual Bundles, cotangent bundle

Picture the vector bundle (E,M, π) as∐p∈MEp =

⋃p∈Mp×Ep. We then define the dual

bundle E? to be the bundle obtained by replacing each fibre Ep by its dual E?p . There is theobvious projection; if v ∈ E?p then π(v) = p.

The transition functions for E?, denoted Adualαβ would be obtained from Aαβ by Adualαβ =

A−Tαβ .

Definition 4.1 The dual bundle of the tangent bundle TM is called the cotangent bundle,and denoted by T ?M.

Definition 4.2 A smooth section of T ?M is called a 1-form.

Locally the tangent bundle has a local frame given by

∂∂xi

i=1,...,n

where x1, ..., xn are local

coordinates. The dual local frame will be denoted dxi.

4.2 Tensor product of bundles

Suppose (E,M, π) and (E,M, π) are two vector bundles of ranks k and l respectively overthe same manifold. The tensor product E ⊗ E is a vector bundle over M of rank kl, whosefibre over p ∈M is the tensor product of the fibres Ep ⊗ Ep.Remark We can also consider Symp(E) or Λp(E)

4.3 Endomorphism Bundle

Definition 4.3 Given a vector bundle E, define End(E) = E? ⊗ E.

One example of this is one that takes a tangent vector on S2 and rotates it anticlockwiseby π

2 .

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4.4 Pull Back Bundles

Imagine vector fields on S1. These are all multiples of ∂∂θ . Now consider u : S1 → R2. We

want to define vector fields along u., i.e. for each p ∈ S1 we want a vector in Tu(p)R2. Suchvector fields along u are considered as sections of the pull back bundle u?(TR2).

Generally suppose u : M → N is a smooth map between manifolds and (E,N , π) is avector bundle. We define the pullback bundle u?(E) to be a vector bundle over M whosefibre at p ∈M is an element of Eu(p).

The total space u?(E) of this is the subspace of M × E of all pairs (x, v) such thatu(x) = π(v). The base space is M. The projections are π : u?(E) → M defined by(x, v) 7→ x. The local trivialisations are given as follows. Given Vα the covering of Nin the definition of E, and local trivialisations ψα : π−1(Vα) → Vα × Rk, consider theopen cover of M by u−1(Vα) =: Vα. Then the new local trivialisations are given byΨα : π−1(Vα)→ Vα×Rk, Ψ(x, v) = (x, Pr2(ψα(v)), where Pr2 is the projection Vα×Rk → Rk

5 Tensors

We used the word tensor to denote an element of ⊗pV ⊗⊗qV ?. The most common situationfor us is when V = TpM

Definition 5.1 A (p, q) tensor field T is a section of the bundle ⊗pTM⊗⊗qT ?M. (p, 0)tensors are often called contravariant and (0, q) tensors are often called covariant.

We can write a tensor field in coordinates (locally) as

T = T i1,...,ipj1,...,jq∂

∂xi1⊗ ...⊗ ∂

∂xip⊗ dxj1 ⊗ ...⊗ dxjq

5.1 Pulling Back covariant tensors

We saw before how to push forward a vector under a map u : M → N , by u? : TpM →Tu(p)N . The dual linear map is given by u? : T ?u(p)N → T ?pM and is defined by

[u?(L)]v = L(u?(v))

for v ∈M.More generally, to pull back a (0, q) tensor

[u?(L1 ⊗ ...⊗ Lq)](v1, ..., vq) = L1(u?(v1))...Lq(u?(vq))

5.2 Differential Forms

Given a manifoldM denote by Λk(M) the bundle of tensors in ⊗kT ?M which are alternat-ing.

Definition 5.2 A (differential) k-form is a section of Λk(M). The space of all suchsections is Ωk(M). There is a convention that Ω0(M) = smooth functions on M

We define the wedge product of two forms as follows. If ω ∈ Ωk(M) and η ∈ Ωl(M) thenthe wedge ω ∧ η ∈ Ωk+l(M) is given by taking the wedge in each fibre.

Locally we can write a k-form as

ω = ai1,...,ikdxi1 ∧ ... ∧ dxik

We can write n forms locally asadx1 ∧ ... ∧ dxn

where a is a smooth function on M

Definition 5.3 A manifold is orientable if there exists a nowhere vanishing n-form.

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5.3 Integration of Differential n-forms

On an orientable manifold we can identify any two nowhere vanishing n-forms ω1, ω2 asfollows. If ω1 = fω2 for some positive function f : M → (0,∞), rather than a negativefunction, then they are equivalent. The two equivalence classes are the two orientationsof M, and an oriented manifold is an orientable manifold with a choice of one of these twoequivalence classes.

Suppose ω = adx1 ∧ ... ∧ dxn is an n form on Rn with compact support. Then we define∫Rn

ω =

∫Rn

adx1...dxn

More generally if ω is an n form on an oriented manifoldM with compact support withina single chart (U, φ), then ∫

Mω = ±

∫φ(U)

adx1...dxn

where x1, ..., xn are coordinates of (U, φ) and ω = adx1 ∧ ... ∧ dxn. Here we use thecoordinate map to map it into the chart. We chose a single chart for simplicity of notation.It is possible if the support isn’t in a single chart.

One takes a + if dx1 ∧ ... ∧ dxn is a positive multiple of the elements of the equivalenceclass, or a − similarly. One can check that this is all well defined.

Note here that this only works for an n-form, where n = dimM. For a k-form we haveno notion of integration on a general manifold.

5.4 Exterior Derivative

We already defined derivatives, also called pushforwards, of maps between manifolds u :M→N . In the special case of functions f :M→ R, consider df as a 1-form: df(X) = X(f).In local coordinates xi, in terms of ∂

∂xi and dxi

df =∂f

∂xidxi

and one can think of this “d” as an operator d : Ω0(M)→ Ω(M). This extends uniquely (itturns out), to d : Ωk(M)→ Ωk+1(M) that is linear and has the properties

1. if ω ∈ Ωk(M) and η ∈ Ωl(M) then

d(ω ∧ η) = (dω) ∧ η + (−1)kω ∧ (dη)

2. d d ≡ 0 or also written d2 = 0.

We call this d the exterior derivative. In coordinates, if

ω = ai1,...,ikdxi1 ∧ ... ∧ dxik

thendω = dai1,...,ik ∧ dx

i1 ∧ ... ∧ dxik

Theorem 5.4 (Stokes’ theorem) If Mn is a compact and oriented manifold, and ω is an(n− 1)-form on M then ∫

Mdω = 0

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6 Connections

A connection gives a way of taking a directional derivative of a section of a vector bundle.

6.1 Motivation

Imagine a vector field V on R2, a point p ∈ R2 and a vector X ∈ TpR2. We write thedirectional derivative of V at p in the direction X as ∇XV .

We would like to generalise this to, for example, differentiating vector fields V on amanifold M, in the direction X ∈ TpM, but we can’t (until we have seen Riemannianmetrics).

The problem is there is no good chart independent definition. To do this differentiationwe need some extra structure on M, that of a connection.

6.2 Definition and elementary properties

Definition 6.1 Let (E,M, π) be a vector bundle over M and Γ(E) be the space of smoothsections. A connection on E is a map ∇ : Γ(TM)× Γ(E)→ Γ(E), (X, v) 7→ (∇Xv) suchthat

1. for all v ∈ Γ(E), the map X 7→ ∇Xv is “ linear over C∞”, namely

∇fX+gY v = f∇Xv + g∇Y v

for all f, g ∈ C∞(M) and X,Y ∈ Γ(TM).

2. for all X ∈ Γ(TM), the map v 7→ ∇Xv is “ linear over R”, namely

∇X(av + bw) = a∇Xv + b∇Xw

for all a, b ∈ R and v, w ∈ Γ(E).

3.∇X(fv) = f∇Xv +X(f)v

for all f ∈ C∞(M) and X ∈ Γ(TM).

It is enough to have a single vector X ∈ TpM and V a section defined on an arbitrarilysmall neighbourhood of p in order to make sense of ∇XV . In particular ∇ : TpM× Γ(E)→Ep. Alternatively we could see ∇ as a map

∇ : Γ(E)→ Γ(T ?M⊗ E)

where in coordinates this is ∇V = dxi ⊗∇ ∂

∂xiV . If X = xj ∂

∂xi∈ Γ(TM) then

dxi ⊗∇ ∂

∂xiV

(xj

∂

∂xj

)= xjdxi(

∂

∂xj)∇ ∂

∂xiV = xjδij∇ ∂

∂xiV = xi∇ ∂

∂xiV = ∇xi ∂

∂xiV = ∇XV

6.3 Extensions of connections to Tensor bundles

Given a connection ∇ on a vector bundle (E,M, π) we can extend it to ⊗pE ⊗ ⊗qE?. Weare also happy when p = 0 = q as then we have ∇ : Γ(TM)× C∞(M)→ C∞(M)

Lemma 6.2 There exists a unique extension of ∇ such that

1. ∇Xf = X(f) for f ∈ C∞(M).

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2. ∇X(T1⊗T2) = (∇XT1)⊗T2 +T1⊗ (∇XT2) for all tensor fields T1, T2 of arbitrary types

3. ∇X(tr(T )) = tr(∇XT ) where T is of type (p, q) with p, q ≥ 1 and tr is the contractionover any pair of E and E? entries.

If we assume that this lemma is true, what would be the extension of ∇ to E?? Forarbitrary L ∈ Γ(E?) and v ∈ Γ(E) and X ∈ Γ(TM) then, for T = L⊗ v we have

X(L(v))1= ∇X(L(v)) = ∇X(tr(L⊗ v))

3= tr(∇x(L⊗ v))

2= tr((∇XL)⊗ v + L⊗ (∇Xv))

3= (∇XL)(v) + L(∇Xv)

and so ∇XL would be determined by (∇XL)(v) = X(L(v)) − L(∇Xv). One could easilyderive an extension to ⊗pE ⊗⊗qE? by condition 2. The existence part of the above lemmareduces to checking the above is a connection. Uniqueness is then clear by construction.

Another way of writing the extended connection is by seeing a tensor as a multilinearmap , eg if T is of type (1, 1), a section of E? ⊗ E we can see it at each point p ∈ M as amultilinear map on Ep ⊗ E?p . If L ∈ Γ(E?) and v ∈ Γ(E) then

(∇XT )(v, L) = X [T (v, L)]− T (∇Xv, L)− T (v,∇XL) (6.1)

6.4 Pullback connection

Recall that given u : M→ N and a vector bundle (E,N , π) we have the pull back bundleu?(E), a vector bundle over M of vectors along u.

If ∇ is a connection on E then a natural pullback connection ∇ is induced on u?(E) aswe now define. Consider v ∈ Γ(E). It induces a section v u for u?(E). Suppose X ∈ TpM,then we wish to write down a formula for ∇X(v u), as follows.

[∇X(v u)](p) = [∇u?(X)v](u(p))

But not all sections of u?(E) are of the form vu, i.e. consider the case that u is the constantmap.

However there is a natural way of extending what we have just defined. Given p ∈ M,choose a local frame ei for E near u(p). Write a general section w ∈ Γ(u?(E)) as w(x) =ai(x)[ei(x) u(x)] as ei u is a local frame for u?(E). By the axioms of connections

∇Xw = ∇X(aiei u) = X(ai)ei u+ ai∇X(ei u)

and the last thing we just made sense of.For example consider γ : (−1, 1) → R2. Then ∇ ∂

∂sw is the directional derivative along

the curve.

6.5 Parallel sections

Definition 6.3 A section v ∈ Γ(E) is parallel if ∇v ≡ 0. We are here seeing ∇ as a mapΓ(E)→ Γ(T ?M⊗ E), so equivalently for all X ∈ TM we have ∇Xv = 0.

For example if E = TR2 then v is parallel if v is parallel. Suppose γ : (−1, 1)→M andconsider w ∈ Γ(γ?(E)). According to the definition, w is parallel if ∇ ∂

∂sw = 0.

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6.6 Parallel Translation

Informally, one would like to push a vector along a curve, keeping it parallel. Formally, thisis defined as:

Suppose γ : (−1, 1)→M and E is a vector bundle overM, with connection ∇. If we aregiven a vector w0 ∈ Tγ(0)M, can we find w ∈ Γ(γ?(E)) such that w(0) = w0 and ∇w ≡ 0?

The answer is yes, and we will call w the parallel translate of w0. This is constructed asfollows.

Observe that γ?(E) is trivial, i.e. we can pick a global frame eα ∈ Γ(γ?(E)). Consider ageneral section w ∈ Γ(γ?(E)) as w(s) = aα(s)eα(s) and then by definition we have

∇ ∂∂sw = ∇ ∂

∂s(aα(s)eα(s))

=∂aα

∂seα + aβ∇ ∂

∂s(eβ)

=∂aα

∂seα + aβΓα1,β(eα)

=

[∂aα

∂s+ aβΓα1,β

]eα

where Γα1,β = θα(∇ ∂∂seβ) is called a Christoffel symbol.

Thus w parallel means the above equation must be zero. This is a system of k firstorder ODEs and the basic theory gives a solution for given initial conditions w0. Beware,that given w0 ∈ Ep for p ∈ M , in general you cannot extend to a parallel section w in aneighbourhood of p.

6.7 Holonomy

A slight variation of the parallel translation construction would involve a map γ : [−1, 1]→M and with specification of w0 ∈ Eγ(−1). We would then find the parallel w ∈ Γ(γ?(E)) asbefore.

Let us consider the special case γ(−1) = γ(1) = p in the case of E = TR2 with ∇ beingthe usual directional derivative, and then parallel translating w0 around γ would give w0 atthe end. However in general we would get back something else.

Consider for exampleM = S2 and ∇ the Levi-Civita connection (to be seen soon). Hereon lines of longitude the vector points in a tangential direction and keeps the same modulusbut changes direction to remain tangential. On equatorial lines it keeps the same directionand modulus. Thus if we prescribe a path from the north pole to the equator, then go rounda quarter of the equator and then back to the north pole on a line of longitude, we haverotated the vector by π/2 In this case, the amount the vector is rotated equals the integral ofthe Gauss curvature K in Ω, the enclosed region. However K ≡ 1 on S2 and so the amountof rotation is Area(Ω)

The phenomenon that w0 is parallel translated to a different vector is called holonomy.This process induces a linear map from Ep → Ep. We will see later that measuring holonomyis measuring curvature.

7 Riemannian Metrics

7.1 Bundle metrics

From Q1.13 in the exercises, on a vector space V , bilinear forms B : V × V → R correspondto elements of B ∈ V ? ⊗ V ?.

A bundle metric on a vector bundle E is an assignment to each fibre Ep of an innerproduct, that is smoothly varying in the following sense.

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Definition 7.1 A bundle metric h on a vector bundle E is a positive definite section ofSym2(E?) ⊂ E? ⊗ E?

Given v, w ∈ Ep we write their inner product as either 〈v, w〉h or h(v, w).Remark All the linear algebra we did with inner products now carries over to this setting,for example we can take traces.

If the vector bundle also has a connection ∇ then we can extend it to a connection onE? ⊗ E? and define

Definition 7.2 ∇ is said to be compatible with a bundle metric h if h is parallel, i.e.∇h ≡ 0.

Equivalently we could have said X(h(v, w)) = h(∇Xv, w) + h(v,∇Xw) for all X ∈ Γ(TM)and v, w ∈ Γ(E) which comes from an equation similar to 6.1.

7.2 A Riemannian metric definition

We here introduce a way of measuring angles and distances on a manifold that will induce aspecial connection; the Levi-Civita connection and curvature.

Definition 7.3 A Riemannian metric g on a manifold M is a bundle metric on TM.

A Riemannian manifold (M, g) is a manifold with Riemannian metric. For example thestandard flat Riemannian metric g on Rn that is given by

g = dx1 ⊗ dx1 + ...+ dxn ⊗ dxn

and note that this is the same as the standard inner product. In other words what this doesis eat two vectors by summing component wise.

Riemannian metrics will allow us to define

1. distance between two points, and length of curves

2. notion of volume

3. a special connection

4. curvature

7.3 Isometries

Definition 7.4 Two Riemannian manifolds (M1, g1) and (M2, g2) are isometric if thereis a diffeomorphism u :M1 →M2 such that u?(g2) = g1.

Riemannian geometry is invariant under isometries

7.4 Induced Metrics

Definition 7.5 Suppose that u :M→ N is an immersion, and g is a Riemannian metricon N . Then the induced metric on M is u?(g)

Observe that the immersion property is to make u?(g) positive definite.

Definition 7.6 An immersion u : (M, g)→ (N , h) is isometric if g = u?h

For example, an isometric immersion γ : (−1, 1) → R2 with the standard metric from lasttime would be a unit speed parameterisation of a curve in R2.

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7.5 Volume forms for Oriented manifolds

On an oriented Riemannian manifoldM, we can define a “ Hodge star” operator as follows:

F : Λk(M)→ Λn−k(M)

for k = 0, 1, ..., n. If θ1, ..., θn is a local dual orthonormal basis for which θ1 ∧ ... ∧ θncorresponds to the orientation, then F(θ1 ∧ ... ∧ θk) = θk+1 ∧ ... ∧ θn

Definition 7.7 Given an oriented Riemannian manifold, we define the (Riemannian) vol-ume form dV ∈ Γ(Λn(M)) by dV = F1

in other words dV = θ1 ∧ ... ∧ θn, with the θi as above.

7.6 Integration on manifolds

We know how to integrate n forms from before. Now we can define integration of a function.

Definition 7.8 Given an oriented Riemannian manifold M and f :M→ R a smooth mapwith compact support, the integral of f is defined to be

∫M fdV .

This definition generalised a lot.

8 Levi-Civita connection

8.1 Definition and Motivation

The Levi-Civita connection is going to be a canonical metric on the tangent bundle of aRiemannian manifold. It is the closest we can get to “directional derivative”.

Suppose we have a connection ∇ on TM . This extends to a connection ∇ on T ?M . givena 1 form ω ∈ Γ(T ?M) there are two ways of differentiating it to get a 2 form:

1. Apply the exterior derivative d

2. Apply ∇ to get a section of T ?M ⊗ T ?M and project onto the alternating tensors andso in all is Alt(∇ω)

The first requirement of the Levi-Civita connection is “torsion free” which means 1 and 2agree (up to normalisation). Here dω = 2Alt(∇ω)

The second requirement of the Levi-Civita connection is that it is compatible with theRiemannian metric.

Theorem 8.1 (Fundamental Theorem of Riemannian Geometry) Given a Rieman-nian manifold (M, g) there exists a unique connection ∇ on TM (the Levi Civita connection)which is both torsion free and compatible with the metric. It is determined by the Koszal for-mula:

〈∇XY, Z〉 =1

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

We first introduce torsion properly. Take ω ∈ Γ(T ?M) and X,Y ∈ Γ(TM). Then

2Alt(∇ω)(X,Y ) = ∇ω(X,Y )−∇ω(Y,X)

= ∇Xω(Y )−∇Y ω(X)

= Xω(Y )− ω(∇XY )− Y ω(X) + ω(∇YX)

= Xω(Y )− Y ω(X)− ω([X,Y ])− ω(∇XY −∇YX − [X,Y ])

From exercise sheet 3, dω(X,Y ) = Xω(Y ) − Y (ω(X)) − ω([X,Y ]), and comparing thesemotivates us to define the torsion as follows:

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Definition 8.2 We define the torsion as

τ(X,Y ) = ∇XY −∇YX − [X,Y ]

and then if τ ≡ 0 we say this is torsion free.

Note that this is C∞(M) linear and so τ is a tensor field. Indeed τ ∈ Γ(⊗2 T ?M⊗TM)

and is even in Γ(Λ2T ?M⊗ TM).There is also an alternative viewpoint. We could instead define

τ := d−Alt ∇ : Γ(T ?M)→ Γ(Λ2T ?M)

but this is equivalent to before because it is also C∞(M) linear and is an element of Γ(TM⊗Λ2T ?M).Proof (of theorem 8.1) First, assume ∇ has the required properties. Then we get fromcompatibility that

〈∇XY,Z〉 = X〈Y,Z〉 − 〈Y,∇XZ〉 (8.1)

and then we use the torsion free condition to write

〈∇XY,Z〉 = X〈Y, Z〉 − 〈Y, [X,Z]〉 − 〈Y,∇ZX〉 (8.2)

and if we cycle X,Y, Z we get the following two equations:

〈∇ZX,Y 〉 = Z〈X,Y 〉 − 〈X, [Z, Y ]〉 − 〈X,∇Y Z〉 (8.3)

〈∇Y Z,X〉 = Y 〈Z,X〉 − 〈Z, [Y,X]〉 − 〈Z,∇XY 〉 (8.4)

Now using (8.4) in (8.3) and the resulting expression in (8.2) we get

〈∇XY,Z〉 = X〈Y,Z〉−〈Y, [X,Z]〉−[Z〈X,Y 〉−〈X, [Z, Y ]〉−[Y 〈Z,X〉−〈Z, [Y,X]〉−〈Z,∇XY 〉]]

and simplifying gives the required equation. Thus if ∇ is a connection with the requiredproperties, then it must be determined by this formula, and so is unique.

Now it is enough to show that ∇ defined by this formula satisfies the connection proper-ties. Q.E.D.

Remark Torsion only makes sense on TM (or T ?M) but compatibility of a connection witha metric makes sense on any vector bundle.Remark The Levi-Civita connection corresponds to directional derivatives on Rn. Take eias the standard basis. Then the connection would be determined by ∇eiej or by 〈∇eiej , ek〉so just need to check that 〈Deiej , ek〉 = 0. Remember that [ei, ej ] = 0 for all i and j.

8.2 Levi-Civita connection and Traces

Just like with a general connection, the L-C connection can be extended to⊗p TM ⊗⊗q T ?M. One of the properties was

∇X(trT ) = tr(∇XT )

where T is a (p, q) tensor field and tr is a contraction over any pair of TM and T ?M entries.But now , with the metric, we can contract over any pair of indices. Thus if T = T ijei ⊗ ejthen trT = gijT

ij = tr13tr24g ⊗ T where tr13 means the contraction of the first and thirdentries of the (2, 2) tensor g ⊗ T . Thus if ∇ is the L-C connection then

∇X(trT ) = ∇X(tr13tr24g ⊗ T )

= tr13tr24∇X(g ⊗ T )

= tr13tr24[(∇Xg)⊗ T + g ⊗ (∇XT )]

= tr13tr24(g ⊗∇XT )

= tr(∇XT )

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and so the L-C connection commutes with any tracesLet u :M→ Rn be an immersion. Define g = u?(gRN ) and so u is an isometric immersion.

We write ∇ for the L-C connection of M. We want to understand ∇XY in terms of ∇Rn.

First, consider u?(Y ), the section of u?(TRn). Let ∇ be the pullback connection of ∇Rn

under u. Then we claim that

u?(∇XY ) = [∇Xu?(Y )]T

where T is projection onto the tangent space. Equivalently

g(∇XY,Z) = 〈∇Xu?(Y ), u?(Z)〉Rn

It is quite common though to confuse X and u?(X) and to write both as X and also to write∇ as ∇Rn

. Thus the above becomes

∇XY = [∇Rn

X Y ]T

in other words the L-C connection is the derivative in Rn projected onto the tangent space.

8.3 Taking more than one derivative; Rough Laplacian

Suppose that (E,M, π) is a vector bundle with connection ∇ andM also has a Riemannianmetric g (which induces the L-C connection). Then we differentiate sections of any tensorproduct of TM, T ?M, E,E? via the product rule. For example it extends to TM⊗ E by

∇X(Y ⊗ V ) = (∇LCX Y )⊗ V + Y ⊗ (∇EXV )

To take two derivatives, if V ∈ Γ(E) then ∇V ∈ Γ(T ?M ⊗ E) and then we can nowdifferentiate again

∇(∇V ) =: ∇2V ∈ Γ(T ?M⊗ T ?M⊗ E)

and we call this the Hessian of V .We can now use the Riemannian metric g to trace over the two T ?M entries in ∇2V

giving us a section of E.

Definition 8.3 The connection Laplacian (or rough Laplacian) of V ∈ Γ(E) is definedto be

∆V := tr12∇2V

Observe that some people take minus this as their Laplacian.Remark If E is of rank zero, sections of E are functions on M and then this Laplacian isthe Laplace-Beltrami operator. With respect to local coordinates xi (this gives a frame

∂∂xi

), and writing gij = g

(∂∂xi, ∂∂xj

), and gij = (dxi, dxj) then

∆LBf := gij∂2f

∂xi∂xj− gijΓkij

∂f

∂xk

8.4 Index notation

This is good for computations where we have lots of derivatives or lots of traces.Suppose ei is a local frame for M and θi is the dual frame and that we have a

Riemannian metric g onM and that ei is orthogonal. Normally upper and lower indices areused to keep track of how objects transform as we change basis. If A transforms objects withupper indices then A−T transforms lower indices. However if we restrict to an orthonormalbasis then we can restrict to A ∈ SO(n), i.e. A = A−T . The upshot is that we only needconsider lower indices.

If ei is an ONB then the dual is also an ONB and then there is an isomorphism V → V ?

given by ei 7→ θi, and so we can write tensors with only lower indices.

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8.4.1 Notation for taking derivatives

If we take one derivative (using the connection ∇) of , say T ∈ Γ(TM⊗ T ?M) then we get∇T ∈ Γ(T ?M⊗ TM⊗ T ?M) and we could write this as

∇iTjkθi ⊗ ej ⊗ θk

This defines ∇iTjk as coefficients. Here we are taking the tensor, applying ∇ and thenexpressing it in terms of the basis, not the other way around.

If we are taking a second derivative, then we often just write

∇i∇jTkl

we could write the rough Laplacian of T as ∇i∇iTjk. However, be careful as writing ∇i∇jis not the same as writing ∇ei∇ej . Think about the following:

∇i∇jf := ∇2f(ei, ej)

∇ei(∇ejf) = ∇ei(ej(f)) = ∇ei(df(ej)) = (∇eidf)(ej) + df(∇eiej) = ∇2f(ei, ej) + df(∇eiej)

and so in the latter we have an extra term.

8.5 Divergence

Definition 8.4 The divergence operator δ : Γ(⊗q+1 T ?M)→ Γ(

⊗q T ?M) is defined by

δT = −tr12∇T

For example, if q = 1 then δT = −∇iTij .In the special case that δ is applied to differential forms, we can show that, for n = dimM,

δ = (−1)nq+1FdF (8.5)

This gives a nice expression for the Laplace-Beltrami operator:

∆LBf = tr∇2f = tr∇df = −δdf = FdFdf (8.6)

A consequence of this is the Divergence theorem.

Theorem 8.5 (Divergence) SupposeM is a compact oriented manifold, and ω ∈ Γ(T ?M).Then ∫

M(δω)dV = 0

Proof ∫M

(δω)dV =

∫M

[(−1)nq+1FdFω]F1

= (−1)nq+1

∫M

F[FdFω]

= ±∫Md(Fω)

Stokes= 0

Q.E.D.

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8.6 Integration by Parts

Lemma 8.6 On an oriented compact manifold (M, g), suppose S ∈ Γ(⊗q T ?M) and T ∈

Γ(⊗q+1 T ?M). Then ∫

M〈δT, S〉 =

∫M〈T,∇S〉

Proof Define the 1-form ωk := Tk,i1,...,iqSi1,...,iq and then δω = 〈δT, S〉 − 〈T,∇S〉 and thenintegrate using the divergence theorem. Q.E.D.

It is best to do the calculation in the proof in index notation. We restrict to the caseq = 1 for simplicity. Then ωk = TkiSi and then

δω : = −∇kωk= −∇k(TkiSi)= −(∇kTki)Si − Tki∇kSi= 〈δT, S〉 − 〈T,∇S〉

Here we use the product rule, the fact that ∇ commutes with traces etc. Without indexnotation, this is a nightmare:

∇Xω = ∇X(tr23T ⊗ S) = tr23∇X(T ⊗ S) = tr23 [∇XT ⊗ S + T ⊗∇XS]

and so∇ω = tr34[∇T ⊗ S] + tr24[T ⊗∇S]

and soδω = −tr∇ω = tr12tr34[∇T ⊗ S] + tr13tr24[T ⊗∇S]

which is what we want.Clearly index notation is better. You could also write the integration by parts formula

in index notation:

−∫M

(∇kTki)SidV =

∫MTki∇kSidV

This formula is maybe more explanative and looks similar to the IBP we knew from secondyear.

9 Geodesics, lengths of paths, Riemannian Distance

9.1 Definition of Geodesic

Consider a map γ from an interval I ⊂ R with standard coordinate s, to a Riemannianmanifold (M, g). Informally γ will be a geodesic if it has zero acceleration.

Definition 9.1 We define the velocity as γ′(s) := γ?(∂∂s

)We write ∇ for the pull-back of the L-C connection ∇ under γ. We adopt the shorthand

Ds = ∇ ∂∂s

Definition 9.2 γ is called a geodesic if Ds(γ′(s)) ≡ 0, or also written ∇ ∂

∂s

(γ?(∂∂s

))= 0

The intuition, as I see it here, is that the velocity is the speed in R pushed forward to lieon the manifold, and so for each point in R we have speed as a vector in the tangent spaceof the manifold.

In other words, γ′(s) is a parallel section of γ?(TM). In the literature it is common tofudge notation and write X = γ′(s) and the equation of the geodesic as ∇XX = 0.

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Theorem 9.3 Given p ∈ M and X ∈ TpM then there exists an open interval I ⊂ Rcontaining 0 and a geodesic γ : I → M with γ(0) = p and γ′(0) = X. Any other geodesicγ : I →M with γ(0) = p and γ′(0) = X agrees with γ on the connected intersection I ∩ I.

The proof of this is essentially writing down the condition to be a geodesic into an ODEand then using existence of ODEs.Proof Pick local coordinates xi near p ∈M and without loss of generality p correspondsto xi = 0 for all i. Write γ(s) = (x1(s), ..., xn(s)). We will write the equation for γ being ageodesic as an ODE for x1, ..., xn. We can write γ′(s) = (x1(s), ..., xn(s)) or more preciselyas γ′(s) = xi(s) ∂

∂xi(γ(s)). Beware, because ∂

∂xi(γ(s)) is not differentiating γ(s). It is the

tangent vector ∂∂xi

at γ(s) ∈M. By definition of the pull-back connection we have

Dsγ′(s) = ∇ ∂

∂s(xi(s)

∂

∂xi)

=∂

∂sxi(s)

∂

∂xi+ xi(s)∇ ∂

∂s

∂

∂xi

= xi(s)∂

∂xi+ xi(s)∇γ′(s)

∂

∂xi

= xk(s)∂

∂xk+ xi(s)xj(s)∇ ∂

∂xj

∂

∂xi

= (xk(s) + xi(s)xj(s)Γkij)∂

∂xk

= 0

and so we have a nice ODE, and so by ODE theory we have a local solution if we specifyxi(0) and xi(0), as we have.

We now prove uniqueness. Suppose the two geodesics are not equal. They must be equalon a neighbourhood of s = 0 by ODE theory. Thus there must be a value s0 > 0 where thegeodesics diverge. At this point, both geodesics have the same velocity, and so by applyingthe ODE argument they agree locally near s0. This is a contradiction. Q.E.D.

9.2 Maximal Geodesics; geodesic completeness

By exploiting the existence and uniqueness of theorem 9.3 we can deduce that there existsa well defined maximal geodesic associated with an arbitrary p ∈ M and X ∈ TpM(associated with an arbitrary X ∈ TM for which we define π(X) = p) i.e. it cannot beextended to any larger interval.

Definition 9.4 The Riemannian manifold (M, g) is geodesically complete if the maxi-mal geodesic associate with each X is defined for all x ∈ R.

As an example, the flat disc is not geodesically complete. Soon (Hopf-Rinow theorem) wewill see that geodesic completeness is equivalent to completeness with respect to a naturalmetric space structure that we will define on a Riemannian manifold.

9.3 Exponential Map

Using the previous section we will define the exponential map as a map from part of TMto M which takes X ∈ TM and maps it to γ(1) where γ is the unique geodesic associatedwith X.

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Definition 9.5 We define DM to be the subset of TM consisting of all X ∈ TM for whichthere is a geodesic associated with X which is defined on an interval I containing [0, 1].

The exponential map exp : DM → M is defined by exp(X) = γ(1) where γ is theunique geodesic associated with X.

We define DM in order to make the exponential map make sense. For example, ifM = S2 then DM = TM.Remark If we restrict to one fibre TpM ∩ DM then we write the exponential map asexpp : TpM∩DM→M

On the sphere expp maps D(0, π), the ball of radius π > 0 in the tangent space in TpM,onto S2 \ south pole.

Lemma 9.6 (Properties of the exponential map.) The following properties hold.

1. DM is an open subset of TM containing all zero length vectors, and each fibre is “starshaped”, i.e. each point can be connected to 0 ∈ TpM by a straight line.

2. Given X ∈ DM, the curve s 7→ exp(sX) is a geodesic defined at least for s ∈ [0, 1] andis the same geodesic γ that defined exp(X) = γ(1) (see Q5.3 Rescalling lemma).

3. exp is a smooth map

We omit the proof. One can either see the Rescaling lemma, or can consider the so called“geodesic flow”.

9.4 Normal Coordinates

Throughout this section, M is a Riemannian manifold.

Lemma 9.7 (Normal Neighbourhood lemma) Given p ∈M there exists ε(p) > 0 suchthat expp is a diffeomorphism from D(0, ε) ⊂ TpM to its image in M.

We will call the supremum of all such ε the injectivity radius at p. Note that this can beinfinity. For example, on the sphere this is π. To prove this lemma we will need the inversefunction theorem for manifolds.Proof (sketch) We want to take the derivative of expp : TpM∩DM→M at 0 ∈ TpM.We get that this is a map

d(expp) : T0(TpM∩DM)→ TpM

but there is a natural identification of T0(TpM) with TpM and so we can write

d(expp) : TpM→ TpM

and by unravelling the definitions we get that

d(expp) = Id

We can show this last line as follows. If we take τ(t) = tX then

d(expp) =d

dt

∣∣∣∣t=0

expp τ(t) =d

dt

∣∣∣∣t=0

expp tX = γX(t)|t=0 = X

where γX is the geodesic at p initially in direction X.In particular it is invertible and so we can apply the inverse function theorem. This then

gives the statement of the lemma. Q.E.D.

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We introduce some terminology. For r ∈ (0, injectivity radius ) we call expp(D(0, r)) ageodesic ball of radius r about p. We call expp(∂D(0, r)) the geodesic sphere about p.

To define normal coordinates pick p ∈M, and choose an orthonormal basis ei of TpM.Define a chart φ by

φ−1(x) = expp(xiei)

where x = (x1, ..., xn) are the coordinates of xiei.

Definition 9.8 The coordinates associated with this chart are called normal coordinates

Normal coordinates xi are very useful, e.g. for making computations because ∇ ∂

∂xi= 0 at

p. Also, gij = δij and so in this case ∂∂xi are orthogonal.

9.4.1 Uniform Normal Neighbourhoods

It is a problem in lemma 9.7 that the injectivity radius could depend horribly on p; it couldbe discontinuous with respect to p.

Lemma 9.9 (Uniform normal neighbourhood lemma) For all p ∈M and any neigh-bourhood U of p there exists a neighbourhood W ⊂ U of p such that there exists a δ > 0 suchthat for all q ∈ W , the restriction of expq to the ball D(0, δ) ⊂ TqM is a diffeomorphismand expq(D(0, δ)) ⊃W .

Proof [Non-examinable?] The key idea here is instead of linearising expp at 0 ∈ TpM welinearise X 7→ (π(X), exp(X)) at 0 ∈ TpM within TM. For a full proof see Lee p78. Q.E.D.

9.5 Adapted Local Frames

We said normal coordinates are good for doing computations. Also useful are adapted localframes. From 3.7 near a point p ∈M there exists a local orthonormal frame ei.

Lemma 9.10 Near any p ∈ M there exists a local orthonormal frame ei such that at p,∇LCei = 0 for all i.

Proof (sketch) Pick any ONB ei of TpM. Extend ei to a neighbourhood of p (anygeodesic ball) by parallel translating each ei radially along geodesics. By Q3.10, this givesan orthonormal frame. By ODE theory, ei is smooth. Q.E.D.

Note we don’t have ∇ei = 0 in a whole neighbourhood of p, in general.

9.6 Lengths of paths/curves

Definition 9.11 The length of a smooth path γ : [a, b]→ (M, g) is defined to be

L(γ) =

∫ b

a|γ′(s)|ds

where |X| = g(X,X)12

Observe that L is independent of the parameterisation, see Q5.6.Remark We can extend this to “ piecewise smooth” curves, i.e. continuous γ : [a, b] →Msuch that there exist finite subdivision of [a, b] where a < a0 < a1 < ... < ak < b such thatγ|[ai−1,ai] is smooth for all i ∈ 1, ..., k. Also allow γ to be constant in which case L(γ) = 0.We call all such curves “admissible”.

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9.7 Riemannian distance

Assume that (M, g) is connected. Given p, q ∈M, define Σp,q to be the set of all admissiblecurves γ : [0, 1]→M such that γ(0) = p and γ(1) = q.

Definition 9.12 The Riemannian distance is a function d :M×M→ [0,∞) defined by

d(p, q) = infγ∈Σp,q

(L(γ))

Observe that this definition implicitly contains the Riemannian metric g. We use Q5.7 tocheck that Σp,q is non empty for arbitrary p, q, if M is connected.

Lemma 9.13 With d defined as above, any connected Riemannian manifold is a metricspace, whose topology agrees with that of the manifold.

9.8 Minimising curve and geodesics

Definition 9.14 An admissible curve γ : [a, b] →M is minimising if L(γ) ≤ L(γ) whereγ is any other admissible curve from γ(a) to γ(b).

Equivalently we could define L(γ) = d(γ(a), γ(b)). Q3.10 gives the fact that geodesics haveconstant speed. Note that we can always parametrise a curve to have constant speed withoutchanging length.

Theorem 9.15 If an admissible curve is minimising and it has constant speed then it is ageodesic. In particular it is smooth, and not just piecewise smooth.

We prove this later.

9.8.1 Variations of Curves

Given γ : [a, b] →M that is smooth we want to consider a variation of γ, more precisely asmooth map F : (−δ, δ)× [a, b]→M such that F (0, ·) = γ. We will consider for η ∈ (−δ, δ)the curves F (η, ·). By smooth here we mean it is smooth on the restriction to (−δ, δ)× (a, b)and the derivatives extend continuously to (−δ, δ)× [a, b] with respect to charts.

9.8.2 First Variation Formula

The goal is to find a formula for ddηL(F (η, ·))|η=0. We will assume that γ = F (0, ·) has unit

speed parametrisation, i.e. |γ′(s)| = 1 for all s ∈ [a, b]. We will use the notation

X =∂F

∂s=∂F

∂s(η, s) = F?

(∂

∂s

)and that

Y =∂F

∂η=∂F

∂η(η, s) = F?

(∂

∂η

)Observe that |X| = 1 when η = 0 and X is the tangent vector along the curves and Y is thevariation vector field.

Theorem 9.16 Given a smooth F as above, for which F (0, ·) has unit speed parametrisation,

d

dηL(F (η, ·))|η=0 = [〈X,Y 〉]s=bs=a −

∫ b

a〈Y,DsX〉ds

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Here the first term on the right hand side describes the change due to the endpoints changing,and the second term the change in shape of the curve.

In the proof of this, we will need the “Symmetry lemma” in Q5.9, which tells us that

DsY = DηX

or equivalently that Ds∂F∂η = Dη

∂F∂s , i.e. in this case the derivatives commute.

Proof By the definition of L, we have

L(F (η, ·)) =

∫ b

a|X|ds =

∫ b

ag(X,X)

12ds

and so using the fact that |X| = 1 we get that

d

dηL(F (η, ·))|η=0 =

∫ b

a

1

2g(X,X)−

12∂

∂ηg(X,X)ds

∣∣∣∣η=0

=

∫ b

ag(DηX,X)ds

∣∣∣∣η=0

=

∫ b

ag(DsY,X)ds

∣∣∣∣η=0

=

∫ b

a

[∂

∂s[g(X,Y )]− g(Y,DsX)

]ds

= [g(X,Y )]s=bs=a −∫ b

ag(Y,DsX)ds

= [〈X,Y 〉]s=bs=a −∫ b

a〈Y,DsX〉ds

Q.E.D.

9.8.3 Proof of Minimising curves are geodesics (Theorem 9.15)

We want to show that if γ : [a, b] → M is a minimising curve with constant speed thenDsγ

′(s) ≡ 0. Let us suppose that Y ∈ Γ(γ?(TM)) with Y (a) = 0 = Y (b). Construct,for sufficiently small δ > 0 a function F : (−δ, δ) × [a, b] → M such that F (0, ·) = γ,F (η, a) = γ(a), F (η, b) = γ(b) and ∂F

∂η (0, ·) = Y , and as usual we set X = ∂F∂s .

By minimising hypothesis we have

d

dηL(F (η, ·))|η=0 = 0

and the first variation formula gives ddηL(F (η, ·))|η=0 = −

∫ ba 〈Y,DsX〉ds due to the end

points not changing. Therefore∫ ba 〈Y,DsX〉ds = 0 for arbitrary Y . Pick a “cut-off” function

φ : R→ [0,∞) such that φ(x) > 0 precisely when x ∈ (a, b), and so it is zero otherwise. SetY (s) = φ(s)DsX and then∫ b

aφ(s)〈DsX,DsX〉ds =

∫ b

aφ(s)|DsX|2ds = 0

and so DsX ≡ 0 i.e. γ is a geodesic since Dsγ′(s) = 0 as X(0, s) = ∂F

∂s (0, s) = γ′(s).

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9.9 Geodesics are locally minimising

Definition 9.17 A curve γ : I →M from some interval I ⊂ R is called locally minimis-ing if for all c ∈ I there exists a neighbourhood U ⊂ I of c such that for all a′, b′ ∈ U thenthe restriction of γ|[a′,b′] is minimising.

Remark If γ is minimising then it is locally minimising.This says that it is locally minimising if it is minimising on a small neighbourhood of the

domain.

Theorem 9.18 Every geodesic is locally minimising.

We proceed to develop the tools to prove this statement

9.9.1 Gauss Lemma

Given p ∈M define the radial distance function on a geodesic ball centred at p by pickingnormal coordinates xi centred at p and setting

r(x) =

(n∑i=1

(xi)2

) 12

This is well defined because it is independent of the choice of orthonormal basis ei fromthe construction of normal coordinates.

Consider the radial geodesic γ : [0, 1] → M such that γ(0) = p, γ(1) = x i.e. γ(s) =expp(sx

iei). Then γ′(0) = xiei and therefore |γ′(s)| = |γ′(0)| = |xiei| = r(x) and thereforeL(γ) = r. We will shortly see that γ is minimising, i.e. d(x, p) = L(γ) = r(x).

Given such an r, it doesn’t immediately make sense to write ∂∂r . Normally this would

mean differentiating with respect to r while keeping other variables fixed. However theseother variables aren’t defined yet.

To define ∂∂r at x in the geodesic ball (not at p), take radial geodesic γ discussed above.

In coordinates γ(s) = (x1, ..., xn) so γ′(s) = xi ∂∂xi|γ(s) which has norm r. We thus define a

unit radial vector field to be∂

∂r=xi

r

∂

∂xi

We could have tried to define this as ∇r = (dr)#, where sharp is the natural isomorphismfrom T ?M to TM. We now want to show that ∂

∂r = ∇r. This will need the Gauss lemma.What we are trying to show is the same as, for all X ∈ TqM q 6= p we need to show

dr(X) = 〈∇r,X〉 = 〈 ∂∂r , X〉. If we set X = ∂∂r then we have the right hand side equal to

RHS =

⟨∂

∂r,∂

∂r

⟩=

∣∣∣∣ ∂∂r∣∣∣∣2 = 1

and

LHS =xi

rdxi(xj

r

∂

∂xj

)=xixj

rδij = 1

and so they are equal. We can decompose a general X as a radial vector and a vector tangentto the geodesic sphere at p through q. Therefore it remains to prove the above for X a tangentvector to the geodesic sphere through q, i.e. for X such that 〈∇r,X〉 = dr(X) = X(r) = 0.Thus we must prove that

〈 ∂∂r,X〉 = 0

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Lemma 9.19 (Gauss) The unit radial vector field ∂∂r is orthogonal to the geodesic spheres.

As a consequence ∇r = ∂∂r and in particular |∇r| = 1.

Proof Let q lie in a geodesic ball centred at p, say on the sphere expp(∂D(0, R)) and writeq = expp(V ) for V ∈ ∂D(0, R).

Let X ∈ TqM be tangent to the geodesic sphere ∂D(0, R) i.e. X(R) = 0. expp isa diffeomorphism on geodesic balls and therefore there exits a W ∈ TV (TpM) such thatd expp(W ) = X. Because X is tangent to geodesic spheres we know that W is tangent to∂D(0, R) at V .

In particular, we can choose a curve σ : (−δ, δ) → ∂D(0, R) such that σ(0) = V andσ′(0) = W . Consider F : (−δ, δ) × [0, 1] →M defined by F (η, s) = expp(sσ(η)). Thereforefor all η ∈ (−δ, δ) then F (η, ·) is a radial geodesic with speed R. The objective is to showthat 1

r∂F∂s (0, 1) is orthogonal to X.

We proceed with a few computations:

∂F

∂η(0, 0) =

d

dη|η=0 expp(0) = 0

∂F

∂η(0, 1) =

d

dη|η=0 expp(σ(η)) = X

by construction. Then

∂

∂s

(⟨∂F

∂η,∂F

∂s

⟩)=

⟨Ds

∂F

∂η,∂F

∂s

⟩+

⟨∂F

∂η,Ds

∂F

∂s

⟩but since F (η, ·) is a geodesic, we have Ds

∂F∂s = 0 and the symmetry lemma (Q5.9) gives

Ds∂F∂η = Dη

∂F∂s and thus we have

∂

∂s

(⟨∂F

∂η,∂F

∂s

⟩)=

⟨Dη

∂F

∂s,∂F

∂s

⟩=

1

2

∂

∂η

⟨∂F

∂s,∂F

∂s

⟩=

1

2

∂

∂η

∣∣∣∣∂F∂s∣∣∣∣2 = 0

since F (η, ·) is a geodesic so ∂F∂s is constant, so its rate of change is zero.

At s = 0 we have ∂F∂η (0, 0) = 0 and therefore⟨

∂F

∂η,∂F

∂s

⟩∣∣∣∣s=0,η=0

= 0

and therefore if we integrate from s = 0 to s = 1 to get, for η = 0,∫ 1

0

∂

∂s

(⟨∂F

∂η,∂F

∂s

⟩)ds =

⟨∂F

∂η,∂F

∂s

⟩∣∣∣∣s=1,η=0

−⟨∂F

∂η,∂F

∂s

⟩∣∣∣∣s=0,η=0

= 0

and so 〈X, ∂F∂s 〉 = 0. Q.E.D.

9.9.2 Radial Geodesics are Initially minimising

Lemma 9.20 Suppose p ∈M and ε ∈ (0,∞) is no more than the injectivity radius. Then forall q ∈ expp(D(0, ε)) (i.e. q = expp(X) for X ∈ D(0, ε)), the radial geodesic s 7→ expp(sX)for s ∈ [0, 1] is a minimising curve from p to q.

Moreover, this is the unique minimising curve from p to q within the class of piecewisesmooth curves from p to q modulo reparametrisation.

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Proof Define R := |X| and consider the radial geodesic with arclength parametrisationγ : s 7→ expp(

sXR ). for s ∈ [0, R]. Observe that |γ′(s)| = 1.

First we need to prove that any other curve from p to q has length at least R, R beingthe length of γ.

We can assume the other curve is smooth (easy to extend to piecewise smooth). Also, bychopping off the start of the curve we may assume it never returns to p (this only decreasesthe length). By chopping off the end of the curve we may assume we have a curve σ :[0, 1]→ expp(D(0, R)) such that σ(0) = p and σ(1) ∈ expp(∂D(0, R)). We will show even σhas length at least R.

We decompose orthogonally to get σ′(s) = a(s) ∂∂r + V (s) for a(s) ∈ R and V (s) ⊥ ∇rand this is for s > 0. Therefore

|σ′(s)|2 = |a(s)|2|∇r|2 + |V (s)|2 ≥ |a(s)|2

Also the inner product of the orthogonal decomposition with ∇r gives

a(s) = 〈σ′(s),∇r〉 = dr(σ′(s))

and so

L(σ) = limε→0

L(σ|[ε,1])

= limε→0

∫ 1

ε|σ′(s)|ds

≥ limε→0

∫ 1

εdr(σ′(s))ds

= limε→0

∫ 1

ε

d

ds[r(σ(s))]ds

= limε→0

[r(σ(1))− r(σ(ε))]

= R− r(σ(0))

= R

It thus remains to show uniqueness. Suppose we have some other curve from p to qwith the least possible length. By the beginning of the proof, our new “competitor” curvehas that it doesnt return to p, and never leaves expp(D(0, ε)) and that σ′(s) = ∂

∂r (if wereparameterise so that |σ′(s)| = 1).

Thus σ and the radial geodesic are integral curves of ∂∂r and they “end” at the q. We

cannot use the beginning as ∂∂r is not defined there. Therefore they are the same, using ODE

theory. Q.E.D.

Corollary 9.21 Within any geodesic ball, the radial distance function r(x) equals the Rie-mannian distance function d(x, p).

Thus geodesic balls expp(D(0, r)) can be written B(p, r) where more generally we defineB(p, r) = x ∈M : d(x, p) < r

9.9.3 Proof that Geodesics are locally minimising

Proof (of theorem 9.18) Given a geodesic curve γ : I →M pick c ∈ I. We need to finda neighbourhood about c where γ is minimising. Without loss of generality we can assumethat I is an open interval.

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We then appeal to lemma 9.9 to find a uniformly normal neighbourhood W about γ(c),i.e. there exists a δ > 0 such that each point of W has a geodesic ball of radius δ thatcontains the whole of W .

Let (a, b) ⊂ I be the connected component of the preimage of W , i.e (γ−1(W )) containingc. We claim that if a < a′ < b′ < b then γ|[a′,b′] is minimising, with c ∈ (a′, b′).

To show this note that γ([a′, b′]) is contained in W by construction and therefore within ageodesic ball of radius δ centred at γ(a′). By construction γ|[a′,b′] is a radial geodesic startingat γ(a′), remaining within the geodesic ball centred at γ(a′). By lemma 9.20, radial geodesicsare minimising while they stay within a geodesic ball. Q.E.D.

9.10 Completeness, Geodesic completeness and the Hopf-Rinow theorem.

Theorem 9.22 (Hopf-Rinow) A connected Riemannian manifold is complete as a metricspace if and only if it is geodesically complete.

In this case, any two points can be connected by a minimising geodesic.

For example, R2 \ 0 is not complete and you cannot connect x and −x by a minimisinggeodesic. We will skip the proof but it can be found in Lee p108.

10 Curvature

We are aiming to talk about the curvature of a Riemannian manifold. We will do this bydefining the curvature of a connection and then we will be able to take the curvature of theLevi-Civita connection.

10.1 Curvature of a Connection

Temporarily we stop assuming that we have a Riemannian metric, at least for this subsection.

Definition 10.1 Given a vector bundle (E,M, π) with connection ∇, the curvature of ∇is the map Γ(TM)× Γ(TM)× Γ(E)→ Γ(E) written (X,Y, v) 7→ R(X,Y )v defined by

R(X,Y )v := −∇X∇Y v +∇Y∇Xv +∇[X,Y ]v

It should be noted that half of everybody takes the opposite sign convention to thatused above. Q6.1 says curvature is not just a map of this form. It is also a tensor. In factR(X,Y ) at p ∈ M only depends on X,Y, v at p and is not dependent on values in someneighbourhood of p. Thus it is in fact a section of T ?M⊗ T ?M⊗ E? ⊗ E. Also notice thesymmetry

R(X,Y )v = −R(Y,X)v

and so R is in fact a section of Λ2T ?M⊗End(E). We can also view R as the endomorphismof E given by v 7→ R(X,Y )v.

R(X,Y )v can be seen as the direction on which v is perturbed if we parallel translate itaround an “infinitesimal loop” in the plane X ∧ Y , see section 6.7, on holonomy.

10.2 Curvature of a Riemannian manifold

Take (M, g) a Riemannian manifold with ∇ the Levi-Civita connection. Then the curvatureR is a section of T ?M⊗T ?M⊗T ?M⊗TM i.e. a (1, 3) tensor field. In this context we canalso use the metric (via musical isomorphism ) to define the (0, 4) tensor Rm(X,Y,W,Z) =〈R(X,Y )W,Z〉. Sometimes we write Rm(g) if we want to emphasise which metric we aretaking the curvature of.

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RemarkRm is invariant under isometries, i.e. if φ is an isometry thenRm(φ?g) = φ?(Rm(g)).All the Riemannian metrics we are looking at are invariant under isometries (by con-

struction).The curvature can be extended to higher order tensors, i.e. R(X,Y )T makes sense for T

any (p, q) tensor field (Q6.2).

Lemma 10.2 Rm has the following symmetries:

1. Rm(X,Y,W,Z) = −Rm(Y,X,W,Z)

2. Rm(X,Y,W,Z) = −Rm(X,Y, Z,W )

3. First Bianchi Identity Rm(X,Y,W,Z) +Rm(Y,W,X,Z) +Rm(W,X, Y, Z) = 0

4. Rm(X,Y,W,Z) = Rm(W,Z,X, Y )

Proof

1. seen before

2.

0 = R(X,Y )〈W,Z〉= R(X,Y )(trW ⊗ Z)

= trR(X,Y )(W ⊗ Z)

= tr[(R(X,Y )W )⊗ Z +W ⊗ (R(X,Y )Z)]

= 〈R(X,Y )W,Z〉+ 〈W,R(X,Y )Z〉

and so Rm(X,Y,W,Z) +Rm(X,Y, Z,W ) = 0

3. Equivalent to R(X,Y )W +R(Y,W )X +R(W,X)Y = 0 which is Q6.3.

4. If we take 3 and permute the entries we get

Rm(X,Y,W,Z) +Rm(Y,W,X,Z) +Rm(W,X, Y, Z) = 0

Rm(Y,W,Z,X) +Rm(W,Z, Y,X) +Rm(Z, Y,W,X) = 0

Rm(W,Z,X, Y ) +Rm(Z,X,W, Y ) +Rm(X,W,Z, Y ) = 0

Rm(Z,X, Y,W ) +Rm(X,Y, Z,W ) +Rm(Y, Z,X,W ) = 0

Add to get cancellation of the first 2 columns and then by 1 and 2 on the final 2columns we get

Rm(W,X, Y, Z)−Rm(Y,Z,W,X) +Rm(W,X, Y, Z)−Rm(Y, Z,W,X) = 0

and the result follows.

Q.E.D.

We also have a differential Bianchi identity.

Lemma 10.3 ∇VRm(W,X, Y, Z) +∇VRm(V,W, Y, Z) +∇WRm(X,V, Y, Z) = 0

The proof of this is Q6.4

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10.3 Sectional Curvature

Definition 10.4 Given p ∈ M take a 2 dimensional subspace π ⊂ TpM. Take an or-thonormal frame X,Y for π. Then the sectional curvature of M at p with respect to πis

K(π) = Rm(X,Y,X, Y )

Note that there is no confusion over the sign of K. It is always the case that spheres havepositive sectional curvature.

If M is a two dimensional manifold then we have to take π = TpM and K is then theGauss curvature at p.

Given K(π) for all p ∈M and π ⊂ TpM we can reconstruct the full curvature Rm.

10.4 Ricci and Scalar curvatures

Sometimes Rm has too much information for our purposes and is hard to compute with, andso we want nicer measures of curvature which still contain enough information.

Definition 10.5 The Ricci curvature Ric : Γ(TM) × Γ(TM) → C∞(M) is the sectionof sym2(T ?M) defined by

Ric(X,Y ) = tr(Rm(X, ·, Y, ·))

In other words Ric(X,Y ) = trZ 7→ R(X,Z)Y =∑

iRm(X, ei, Y, ei) for ei an orthonor-mal frame field.

Definition 10.6 The scalar curvature is defined to be R ∈ C∞(M) by

R = trg(Ric)

In other words R = gijRij where Rijdxi ⊗ dxj = Ric

Remark Ric contains just enough information to be useful:

• It contains the volume growth

• If ∆ represents the Laplace -Beltrami operator, then in normal coordinates ∆gij =−2

3Rij

In two dimensions, the scalar curvature R is twice the Gauss curvature.We consider the second Bianchi identity again:

∇VRm(W,X, Y, Z) +∇VRm(V,W, Y, Z) +∇WRm(X,V, Y, Z) = 0

If we trace over W and Y and then over X and Z we get

∇VR+ δRic(V ) + δRic(V ) = 0

we summarise this as follows.

Lemma 10.7 (Contracted second Bianchi identity)

dR+ 2δRic = 0

Identities such as these reflect the fact that the whole subject is invariant under pullback.

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10.5 Einstein Metrics

An Einstein metric is a metric g such that

Ric = λg

for some λ ∈ R. Some people would define Einstein metrics to be those satisfying this forλ :M→ R, i.e. by taking trace of this giving R = λn. Then we have

Ric =R

ng

Lemma 10.8 (Schur) The two above definitions of an Einstein metric are the same, solong as dimM 6= 2

Proof We take the divergence to get

δRic = δ

(R

ng

)=

1

nδ(Rg) = − 1

ntr(∇Rg) = −dR

n

Then using lemma 10.7

dR = −nδRic =n

2dR

and if n = 2 this is vacuous. If n 6= 2 then dR = 0 and so R is constant Q.E.D.

10.6 Curvature Operator

Given the symmetries of Rm (1 and 2 in lemma 10.2) we can define a linear map

R : Λ2(TM)→ Λ2(TM)

called the curvature operator, defined by

〈R(X ∧ Y ),W ∧ Z〉 = Rm(X,Y,W,Z)

By symmetry 4 of lemma 10.2 we have that R is symmetric. Therefore we can diagonaliseR. If all the eigenvalues are positive we say the manifold has positive curvature operator.Remark (non exam) A common theme in Riemannian geometry is that the curvature con-dition on a manifold implies a particular topological condition on the manifold. For example,by Gauss Bonnet , a compact surface with positive Gauss curvature must be S2 or RP 2.

Theorem 10.9 (Bohn-Wilking 2006) (non exam) A simply connected compact manifoldwith positive curvature operator must be diffeomorphic to Sn.

10.7 Curvature in Index notation

We introduce several conventions for the coefficients of curvature with respect to coordinates:

Rm = Rijkldxi ⊗ dxj ⊗ dxk ⊗ dxl

Ric = Rijdxi ⊗ dxj

and therefore Rij = gklRikjl and R = gijRij = gijgklRikjl.Working with respect to an orthonormal basis, or covector field, we have Rij = Rikjk and

R = Rii = Rikik and we often use this notation in calculations. As a first example lets redothe contracted second Bianchi. This is

∇iRjklm +∇kRijlm +∇jRkilm = 0

and tracing over j and l and k and m gives

∇iR−∇kRik −∇jRij = 0

which is what we want in index notation.

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11 Conquering Calculation

11.1 Switching Derivations

Given some general tensor T , if we apply ∇ twice to get ∇2T ∈ Γ(T ?M⊗T ?M⊗...) and thenapply first T ?M⊗ T ?M part to X,Y ∈ Γ(TM) we write the result ∇2

X,Y T . For example if

T is a 1-form then ∇2X,Y T := ∇2T (X,Y )

∇2X,Y T = (∇X∇T )(Y )−∇X(∇T (Y ))−∇T (∇XY ) = ∇X∇Y T −∇∇XY T

and as a consequence

−∇2X,Y +∇2

Y,X = −(∇X∇Y −∇∇XY ) + (∇Y∇X −∇∇YX)

If it is torsion free then this is equal to

−∇X∇Y +∇Y∇X +∇[X,Y ] = R(X,Y )

so if we commute two derivatives during a computation, we pick up an extra “curvature”term.

In practise we often want to write the curvature tensor in terms of R(X,Y ) acting on avector field.

For example suppose w is a 1-form, using R(X,Y )f = 0 and the Leibniz rule we have

0 = R(X,Y )(w(Z)) = [R(X,Y )w](Z) + w[R(X,Y )Z]

and this leads to the Ricci identity

(−∇2X,Y w +∇2

Y,Xw)(Z) = −w[R(X,Y )(Z)]

i.e. the way we switch derivatives of a 1-form. More generally, for any A ∈ Γ(⊗kT ?M) wehave

(−∇2X,YA+∇2

Y,XA)(w, z, ...) = [R(X,Y )A](w, z, ...) =

= −A(R(X,Y )w, z, ...)−A(w,R(X,Y )z, ...)− ...

but this is more useful in index notation. This is with respect to an orthonormal frame ei.Since ∇2

X,Y w = ∇2w(X,Y ) we have this equal to

∇2X,Y w = (∇i∇jwk)XiY jek

and alsoR(X,Y )w = −∇i∇jwkXiY jek +∇j∇iwkXiY jek

Also R(X,Y )w = RijklXiY jwkel and we would normally then write that

Rijklwk = −∇i∇jwk +∇j∇iwk

or∇i∇jwk = ∇j∇iwk +Rijklwl

More generally we have

∇i∇jAk1...km = ∇j∇iAk1...km +RijklAlk2...km +RijklAk1lk3...km + ...

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11.2 Bochnaw/Weitzenbock formula

The spirit of this application is that a manifold with positive Ricci curvature tells us stuffabout the topology.

Proposition 11.1 Suppose f :M→ R with M a Riemannian manifold. Then

1

2∆|df |2 = |Hess(f)|2 + 〈df, d|∆LBf |〉+ Ric(∇f,∇f)

We make some clarification

1. Hess(f) = ∇(df) ∈ Γ(T ?M⊗ T ?M) so we can talk about its norm

2. another way of writing 〈df, dh〉 would be 〈∇f,∇h〉

3. In index notation 〈df, dh〉 = (∇if)(∇ih)

4. In index notation ∆LBf = ∇i∇if

5. In index notation |Hess(f)|2 = (∇i∇jf)(∇i∇jf)

6. Hess is symmetric i.e. ∇i∇jf = ∇j∇if

7. Ricci Identity ∇i∇iwk = ∇j∇iwk +Rijklwl

Proof

1

2∆|df |2 =

1

2∇i∇j((∇jf)(∇jf))

= ∇i[(∇i∇jf)(∇jf)]

= (∇i(∇i∇jf))(∇jf) + (∇i∇jf)(∇i∇jf)

= (∇i∇j∇if)(∇jf) + |Hess(f)|2

= (∇j(∆LBf))(∇jf) +Rjl(∇lf)(∇jf) + |Hess(f)|2

= 〈d|∆LBf |, df〉+ Ric(∇f,∇f) + |Hess(f)|2

Q.E.D.

12 Second Variation Formula

In section 8.8.2 we computed the first variation formula for the length of a path. We have thefollowing basic set up. Let γ : [a, b] →M be a smooth curve, and F : (−δ, δ) × [a, b] →Mis the variation and consider the length L(F (η, ·)). Previously we assumed that γ had unitspeed. Now we will assume that it is a geodesic. Also assume that the end points are keptfixed i.e.

F (η, a) = γ(a), F (η, b) = γ(b)

for all η ∈ (−δ, δ).

Definition 12.1 Define the normal projection of Y to be

Y ⊥ = Y − 〈Y, γ′(s)〉γ′(s)

in other words this is Y minus the part parallel to γ′.Suppose that X = ∂F

∂s and Y = ∂F∂η , and note that |X| = 1 for η = 0.

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Theorem 12.2 Given F a smooth curve as above with F (0, ·) a unit speed geodesic and withfixed endpoints, we have

d2

dη2

∣∣∣∣η=0

L(F (η, ·)) =

∫ b

a

[|DsY

⊥|2 −Rm(Y ⊥, γ′(s), Y ⊥, γ′(s))]ds

Note that the second derivative of length on a negatively curved manifold is positive, and soall geodesics are stable, i.e. if one starts on one, then one stays on one.Proof Start by taking a first derivative of L(F (η, ·)). This is a bit like the computation forfirst variation formula except we cannot yet set η = 0. Recall that, from the definition,

L(F (η, ·)) =

∫ b

a

∣∣∣∣∂F∂s∣∣∣∣ ds =

∫ b

ag(X,X)

12ds

and also observe that ddη [g(X,X)] = 2g(DηX,X) = 2g(DsY,X) by the symmetry lemma.

Thus

d

dηL(F (η, ·)) =

∫ b

a

1

2g(X,X)−

12d

dη(g(X,X))ds =

∫ b

ag(X,X)−

12 g(DsY,X)ds

Differentiate with respect to η again and now set η = 0 to get

d2

dη2

∣∣∣∣η=0

L(F (η, ·)) =

∫ b

a−1

2g(X,X)−

32

[d

dηg(X,X)

]g(DsY,X) + g(X,X)−

12×

× (g(DηDsY,X) + g(DsY,DηX))ds

=

∫ b

a−1

2

[d

dηg(X,X)

]g(DsY,X) + g(DηDsY,X) + g(DsY,DηX)ds

=

∫ b

a−[g(DsY,X)]2 + g(DηDsY,X) + g(DsY,DsY )ds

=

∫ b

a−[g(DsY,X)]2 + |DsY |2 + g(DsDηY,X) + g(R(

∂

∂s,∂

∂η)Y,X)ds

=

∫ b

a−[g(DsY,X)]2 + |DsY |2 +

∂

∂s[g(DηY,X)]

− g(DηY,DsX) + g(R(X,Y )Y,X)ds

=

∫ b

a−[g(DsY,X)]2 + |DsY |2 +Rm(X,Y, Y,X)ds

since DsX = 0 for η = 0 as we are on a geodesic, and the other term is zero as it dependson the change of the endpoints, which is zero.

Now Y = Y ⊥ + Y T with Y T = 〈Y,X〉X and so Ds(YT ) = 〈DsY,X〉X as DsX = 0 for

η = 0. Thus|(DsY )T |2 = |〈DsY,X〉X|2 = 〈DsY,X〉2

and also

(DsY )⊥ = DsY − 〈DsY,X〉X = Ds(Y⊥) +Ds(Y

T )− 〈DsY,X〉X = Ds(Y⊥)

and thus|DsY |2 = |(DsY )⊥|2 + |(DsY )T |2 = |DsY

⊥|2 + g(DsY,X)2

We then conclude that

d2

dη2|η=0L(F (η, ·)) =

∫ b

a|DsY

⊥|2 +Rm(X,Y Y,X)ds =

∫ b

a|DsY

⊥|2 +Rm(Y, γ′(s), Y, γ′(s))ds

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Note that Rm(X,Y,X, Y ) = Rm(X,Y ⊥, X, Y ⊥) because by decomposing, we have

Rm(X,X, ·, ·) = 0 Rm(·, ·, X,X) = 0

because of the symmetries of Rm. Q.E.D.

Remark This formula works because γ is assumed to be a geodesic. This is why it onlydepends on Y at η = 0. Analogous fact in finite dimensions is this. If σ(η) is a smooth curvein R2 with σ(0) = 0 and L : R2 → R a smooth function then

d2

dη2

∣∣∣∣η=0

L(σ(η)) = (HessL)(σ′(0), σ′(0))

provided ∇L = 0 at 0. Otherwise we would get a σ′′ term.A consequence of this is given a minimising geodesic γ : [a, b] → M then for any Y ∈

Γ(γ?(TM)) with Y (a) = 0 = Y (b) and Y (s) ⊥ γ′(s) for all s ∈ [a, b]. Define F : (−δ, δ) ×[a, b] → M by F (η, s) = expγ(s)(ηY ) and apply the second variation formula. Then γ

minimising implies d2

dη2|η=0L(F (η, ·)) ≥ 0 and therefore we have the following.

Corollary 12.3 Under the circumstances above

Q(Y ) :=

∫ b

a|DsY |2 −Rm(Y, γ′(s), Y, γ′(s))ds ≥ 0

13 Classical Riemannian Geometry Results: Bonnet-Myers

Definition 13.1 The diameter of a Riemannian connected manifold is defined to be

diam(M, g) = supp,q∈M

dg(p, q) ∈ (0,∞]

Theorem 13.2 (Myers) Suppose r > 0 and (M, g) is a complete connected Rieman-nian manifold of dimension n such that Ric ≥ n−1

r2g i.e. if X ∈ TM then Ric(X,X) ≥

n−1r2g(X,X). Then M is compact and

diam(M, g) ≤ πr

Thus positive curvature forces the manifold to close in on itself.Remark This is sharp for the sphere, e.g. a sphere of radius r has Ric = n−1

r2g.

Proof Start by proving the diameter estimate. Pick arbitrary points p, q ∈ M and defineL = dg(p, q). We want to prove that L ≤ πr.

The Hopf-Rinow theorem tells us that there exists a minimising geodesic γ : [0, L]→Msuch that γ(0) = p and γ(L) = q, and with unit speed. Pick an orthonormal basis ei of TpMwith en = γ′(0). Parallel translate this frame ei along γ to give sections ei ∈ Γ(γ?(TM))and note that for all s ∈ [0, L] we have ei(s) is an orthonormal basis of Tγ(s)M. Note forall s that en(s) = γ′(s) because en(0) = γ′(0) by construction and both en(s) and γ′(s) areparallel (satisfy same ODEs).

Define for all i ∈ 1, ..., n− 1

Yi(s) = sin(πsL

)ei(s)

and observe that Yi(0) = 0 = Yi(L) and that Yi(s) ⊥ γ′(s).Apply corollary 12.3 to get that

Q(Yi) =

∫ L

0|DsYi|2 −Rm(Yi, γ

′(s), Yi, γ′(s))ds ≥ 0

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but DsYi = πL cos

(πsL

)ei(s) + sin

(πsL

)× 0 = π

L cos(πsL

)ei(s) and therefore

|DsYi|2 =π2

L2cos2

(πsL

)and also

Rm(Yi, γ′(s), Yi, γ

′(s)) = sin2(πsL

)Rm(ei, γ

′(s), ei, γ′(s))

therefore

0 ≤∫ L

0

π2

L2cos2

(πsL

)− sin2

(πsL

)Rm(ei, γ

′(s), ei, γ′(s))ds

and then sum over i = 1, ..., n− 1 and note that Rm(en, γ′(s), en, γ

′(s)) = 0 and so

0 ≤∫ L

0

π2(n− 1)

L2cos2

(πsL

)− sin2

(πsL

)Ric(γ′(s), γ′(s))ds

but Ric(γ′(s), γ′(s)) ≥ n−1r2g(γ′(s), γ′(s)) = n−1

r2and therefore

0 ≤∫ L

0

π2(n− 1)

L2cos2

(πsL

)− n− 1

r2sin2

(πsL

)ds

and so

0 ≤ π2(n− 1)

L2− n− 1

r2

i.e. L2 ≤ π2r2 and so L ≤ πr as L > 0.To show that M is compact, note that by the diameter estimate, M is the image of

D(0, πr) under expp. The image of a compact set under a continuous map is compact, so wehave the result. Q.E.D.

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