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 by
Peter Topping, typeset by Matthew Egginton.
No guarantee is given that they are accurate or applicable, but hopefully they will assist
your study.
Please report any errors, factual or typographical, to m.egginton@warwick.ac.uk
1
MA4C0 Differential Geometry Lecture Notes Autumn 2012
1
Introduction
The following is a very sketchy background in certain parts of the course, included to give
some 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 a
neighbourhood that is homeomorphic to a ball in Rn .
Some examples of manifolds are R, S 1 , S 2 , T 2 , RP n , SO(n). An example that isn’t a
manifold is where three half lines meet at the same point. The point is where it fails, as at
this point it is not homeomorphic to a line.
1.1
1.1.1
Intrinsic Curvature of a Surface (Gauss Curvature)
Geodesic Flow
Curvature is very closely connected with the stability of geodesic flow. A positive curvature
encourages 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 same
function in Rn we can measure curvature, for example if the manifold is positively curved
then 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 curvature
then
Z
KdV = 4π(1 − g)
M
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 Riemannian metric with constant curvature
• +1 if M is the sphere
• 0 if g = 1
• -1 if g ≥ 2
Theorem 1.4 (Cartan -Hadamard) Suppose M is a Riemannian manifold which is simply 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 sectional
curvature 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 − K6 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 space
which is second countable, together with an open cover {Uα }α∈I of M and maps φα : Uα → Rn
homeomorphic onto their image such that whenever Uα ∩Uβ 6= ∅ then φα ◦φ−1
β : φβ (Uα ∩Uβ ) →
Rn is smooth
The φα or (Uα , φα ) are called charts, and φα ◦ φ−1
β 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 have
f ◦ φ−1
:
φ
(U
α
α ) → R is smooth as a function R → R.
α
cβ ◦ F ◦ φ−1 is
c between manifolds is smooth if for any charts φ
2. A map F : M → M
α
smooth.
c is a diffeomorphism if it is smooth and also a bijection with
3. A map F : M → M
smooth 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
of
Pnthe space
∂f
smooth functions in a neighbourhood of p which can be written locally as f → i=1 ai ∂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 map
u : 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). The
tangent space Tp M is the space of all vectors at p. This linear map is normally written as
du : Tp M → Tu(p) N .
Definition 2.5 u : M → N is an immersion if for all p ∈ M we have that du : Tp M →
Tu(p) N is injective. It is a submersion if du is surjective for all p and it is an embedding
if 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 all
pairs (p, X). This will be a 2n dimensional manifold where n = dim M, and will have some
extra structure; that of a vector bundle.
Definition 2.6 A tangent vector field is an assignment to each point p ∈ M of a vector
∂
for smooth local coefficients ai (x).
in Tp M, which can be written ai (x) ∂x
i
Definition 2.7 A smooth real vector bundle (E, M, π) of rank k ∈ N0 is a smooth manifold E of dimension m + k, a smooth manifold M of dimension m and a smooth surjective
map 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 ◦ ψβ−1 : (Uα ∩ Uβ ) × Rk → (Uα ∩ Uβ ) × Rk takes the
form (x, v) 7→ (x, Aαβ v) 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 space
structure.
π
We often write E → M instead of the triple.
Definition 2.9 A smooth bundle morphism (map) H from (E, M, π) to (E 0 , M0 , π 0 ) is
a smooth map H : E → E 0 which maps Ep to some Ep0 0 as a linear map
Definition 2.10 Two bundles (E, M, π) and (E 0 , M 0 , π 0 ) are isomorphic if there is a
smooth bundle map H : (E, M, π) → (E 0 , M0 , π 0 ) which has an inverse that is a smooth
bundle map
Remark We only consider vector bundles up to isomorphism.
The tangent bundle is a vector bundle of rank m = dim M. Given a manifold M with
charts (Uα , φα ) then the total space T M is the space of pairs (p, X) with p ∈ M and X ∈
Tp M and π(p, X) = p and the local trivialisations ψα are defined by ψα : π −1 (Uα ) → Uα ×Rm
∂
) 7→ (p, a1 , ..., am )
given by (p, ai (x) ∂x
i
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. We
write bases of V as {ei }, with a lower index. We write a general element as ai ei , with an
upper index on the coefficients. We write bases of V ? as {θi }, with upper indices, and a
general element as bi θi . To understand why, see question 1.10 in problems.
3.2
Inner products and dual spaces
We can equip V with an inner product h, i. This then induces an isomorphism Θ : V → V ?
by [Θ(v)](w) = hv, wi for v, w ∈ V . By Θ(v) we mean an element in the dual space, nothing
more. This allows us to extend the inner product to V ? by
hΘ(v), Θ(w)i := hv, wi
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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 of
V × 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. In
practise this means that V ⊗ W can be considered as all finite linear combinations of v ⊗ w
where we agree that (λv) ⊗ w = λv ⊗ w = v ⊗ (λw) etc following the above. If {ei }i is a basis
for V and {fj }j∈J is a basis for W then {ei ⊗ej }I,J is a basis for V ⊗W . However, one should
note 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 = Tp M. Such a tensor
can be written as
i ,...,i
Tj11,...,jqp ei1 ⊗ ... ⊗ eip ⊗ θj1 ⊗ ... ⊗ θjq
and for brevity we often only write the coefficients T .
3.4
Extending inner product to tensors
Given h, i on V we can define an inner product on V ⊗ ... ⊗ V ⊗ V ? ⊗ ... ⊗ V ? by
hv1 ⊗ ... ⊗ vp ⊗ L1 ⊗ ... ⊗ Lq , v˜1 ⊗ ... ⊗ v˜p ⊗ L˜1 ⊗ ... ⊗ L˜q i = hv1 , v˜1 i...hvp , v˜p ihL1 , L˜1 i...hLq , L˜q i
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 taking
each term L ⊗ v and replace with L(v), for example Tij θj ⊗ ei then the contraction is Tii .
Given an element of V ⊗ V (or V ⊗ V ) we can use the isomorphism Θ : V → V ? to turn it
into 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 we
switch the entries of each term. Also T = T ij ei ⊗ ej is symmetric if T ij is a symmetric
matrix. More generally a (p, 0) tensor is symmetric if it is invariant when we switch any
two 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 if
switching any two entries reverses the sign. The vector space of all such tensors is denoted
Λp V
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→
1
2 (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→
1
2 (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
`
S
Picture the vector bundle (E, M, π) as p∈M Ep = p∈M {p} × Ep . We then define the dual
bundle E ? to be the bundle obtained by replacing each fibre Ep by its dual Ep? . There is the
obvious projection; if v ∈ Ep? then π(v) = p.
dual
The transition functions for E ? , denoted Adual
αβ would be obtained from Aαβ by Aαβ =
−T
Aαβ .
Definition 4.1 The dual bundle of the tangent bundle T M 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 (Ẽ, M, π̃) are two vector bundles of ranks k and l respectively over
the same manifold. The tensor product E ⊗ Ẽ is a vector bundle over M of rank kl, whose
fibre over p ∈ M is the tensor product of the fibres Ep ⊗ Ẽp .
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 S 2 and rotates it anticlockwise
by
π
2.
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4.4
Pull Back Bundles
∂
. Now consider u : S 1 → R2 . We
Imagine vector fields on S 1 . These are all multiples of ∂θ
1
want to define vector fields along u., i.e. for each p ∈ S we want a vector in Tu(p) R2 . Such
vector fields along u are considered as sections of the pull back bundle u? (T R2 ).
Generally suppose u : M → N is a smooth map between manifolds and (E, N , π) is a
vector bundle. We define the pullback bundle u? (E) to be a vector bundle over M whose
fibre 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 that
u(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 N
in the definition of E, and local trivialisations ψα : π −1 (Vα ) → Vα × Rk , consider the
open cover of M by {u−1 (Vα )} =: {V˜α }. Then the new local trivialisations are given by
Ψα : π −1 (V˜α ) → V˜α ×Rk , Ψ(x, v) = (x, P r2 (ψα (v)), where P r2 is the projection Vα ×Rk → Rk
5
Tensors
We used the word tensor to denote an element of ⊗p V ⊗ ⊗q V ? . The most common situation
for us is when V = Tp M
Definition 5.1 A (p, q) tensor field T is a section of the bundle ⊗p T M ⊗ ⊗q T ? 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 ,...,ip j1 ,...,jq
5.1
∂
∂
⊗ ... ⊗ ip ⊗ dxj1 ⊗ ... ⊗ dxjq
i
1
∂x
∂x
Pulling Back covariant tensors
We saw before how to push forward a vector under a map u : M → N , by u? : Tp M →
? N → T ? M and is defined by
Tu(p) N . The dual linear map is given by u? : Tu(p)
p
[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 manifold M denote by Λk (M) the bundle of tensors in ⊗k T ? M which are alternating.
Definition 5.2 A (differential) k-form is a section of Λk (M). The space of all such
sections 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) then
the 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 ,...,ik dxi1 ∧ ... ∧ dxik
We can write n forms locally as
adx1 ∧ ... ∧ 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 as
follows. If ω1 = f ω2 for some positive function f : M → (0, ∞), rather than a negative
function, then they are equivalent. The two equivalence classes are the two orientations
of M, and an oriented manifold is an orientable manifold with a choice of one of these two
equivalence classes.
Suppose ω = adx1 ∧ ... ∧ dxn is an n form on Rn with compact support. Then we define
Z
Z
adx1 ...dxn
ω=
Rn
Rn
More generally if ω is an n form on an oriented manifold M with compact support within
a single chart (U, φ), then
Z
Z
adx1 ...dxn
ω=±
M
φ(U )
{x1 , ..., xn }
where
are coordinates of (U, φ) and ω = adx1 ∧ ... ∧ dxn . Here we use the
coordinate 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 equivalence
class, or a − similarly. One can check that this is all well defined.
Note here that this only works for an n-form, where n = dim M. For a k-form we have
no 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 ).
∂
i
In local coordinates {xi }, in terms of { ∂x
i } and {dx }
df =
∂f i
dx
∂xi
and one can think of this “d” as an operator d : Ω0 (M) → Ω(M). This extends uniquely (it
turns 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)k ω ∧ (dη)
2. d ◦ d ≡ 0 or also written d2 = 0.
We call this d the exterior derivative. In coordinates, if
ω = ai1 ,...,ik dxi1 ∧ ... ∧ dxik
then
dω = dai1 ,...,ik ∧ dxi1 ∧ ... ∧ dxik
Theorem 5.4 (Stokes’ theorem) If Mn is a compact and oriented manifold, and ω is an
(n − 1)-form on M then
Z
dω = 0
M
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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 ∈ Tp R2 . We write the
directional derivative of V at p in the direction X as ∇X V .
We would like to generalise this to, for example, differentiating vector fields V on a
manifold M, in the direction X ∈ Tp M, but we can’t (until we have seen Riemannian
metrics).
The problem is there is no good chart independent definition. To do this differentiation
we 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 smooth
sections. A connection on E is a map ∇ : Γ(T M) × Γ(E) → Γ(E), (X, v) 7→ (∇X v) such
that
1. for all v ∈ Γ(E), the map X 7→ ∇X v is “ linear over C ∞ ”, namely
∇f X+gY v = f ∇X v + g∇Y v
for all f, g ∈ C ∞ (M) and X, Y ∈ Γ(T M).
2. for all X ∈ Γ(T M), the map v 7→ ∇X v is “ linear over R”, namely
∇X (av + bw) = a∇X v + b∇X w
for all a, b ∈ R and v, w ∈ Γ(E).
3.
∇X (f v) = f ∇X v + X(f )v
for all f ∈ C ∞ (M) and X ∈ Γ(T M).
It is enough to have a single vector X ∈ Tp M and V a section defined on an arbitrarily
small neighbourhood of p in order to make sense of ∇X V . In particular ∇ : Tp M × Γ(E) →
Ep . Alternatively we could see ∇ as a map
∇ : Γ(E) → Γ(T ? M ⊗ E)
where in coordinates this is ∇V = dxi ⊗ ∇
∂
∂xi
∂
V . If X = xj ∂x
i ∈ Γ(T M) then
∂
j ∂
dx ⊗ ∇ ∂ V x
= xj dxi ( j )∇ ∂ V = xj δji ∇ ∂ V = xi ∇ ∂ V = ∇xi ∂ V = ∇X V
j
i
∂x
∂x
∂x
∂xi
∂xi
∂xi
∂xi
i
6.3
Extensions of connections to Tensor bundles
Given a connection ∇ on a vector bundle (E, M, π) we can extend it to ⊗p E ⊗ ⊗q E ? . We
are also happy when p = 0 = q as then we have ∇ : Γ(T M) × C ∞ (M) → C ∞ (M)
Lemma 6.2 There exists a unique extension of ∇ such that
1. ∇X f = X(f ) for f ∈ C ∞ (M).
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2. ∇X (T1 ⊗ T2 ) = (∇X T1 ) ⊗ T2 + T1 ⊗ (∇X T2 ) for all tensor fields T1 , T2 of arbitrary types
3. ∇X (tr(T )) = tr(∇X T ) where T is of type (p, q) with p, q ≥ 1 and tr is the contraction
over any pair of E and E ? entries.
If we assume that this lemma is true, what would be the extension of ∇ to E ? ? For
arbitrary L ∈ Γ(E ? ) and v ∈ Γ(E) and X ∈ Γ(T M) then, for T = L ⊗ v we have
1
3
X(L(v)) = ∇X (L(v)) = ∇X (tr(L ⊗ v)) = tr(∇x (L ⊗ v))
2
3
= tr((∇X L) ⊗ v + L ⊗ (∇X v)) = (∇X L)(v) + L(∇X v)
and so ∇X L would be determined by (∇X L)(v) = X(L(v)) − L(∇X v). One could easily
derive an extension to ⊗p E ⊗ ⊗q E ? by condition 2. The existence part of the above lemma
reduces 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 multilinear
map , eg if T is of type (1, 1), a section of E ? ⊗ E we can see it at each point p ∈ M as a
multilinear map on Ep ⊗ Ep? . If L ∈ Γ(E ? ) and v ∈ Γ(E) then
(∇X T )(v, L) = X [T (v, L)] − T (∇X v, L) − T (v, ∇X L)
6.4
(6.1)
Pullback connection
Recall that given u : M → N and a vector bundle (E, N , π) we have the pull back bundle
u? (E), a vector bundle over M of vectors along u.
˜ is induced on u? (E) as
If ∇ is a connection on E then a natural pullback connection ∇
?
we now define. Consider v ∈ Γ(E). It induces a section v ◦ u for u (E). Suppose X ∈ Tp M,
˜ X (v ◦ u), as follows.
then we wish to write down a formula for ∇
˜ X (v ◦ u)](p) = [∇u (X) v](u(p))
[∇
?
But not all sections of u? (E) are of the form v ◦u, i.e. consider the case that u is the constant
map.
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 (ai ei ◦ u) = X(ai )ei ◦ u + ai ∇
˜ X (ei ◦ u)
∇
and the last thing we just made sense of.
˜ ∂ w is the directional derivative along
For example consider γ : (−1, 1) → R2 . Then ∇
∂s
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 map
Γ(E) → Γ(T ? M ⊗ E), so equivalently for all X ∈ T M we have ∇X v = 0.
For example if E = T R2 then v is parallel if v is parallel. Suppose γ : (−1, 1) → M and
˜ ∂ w = 0.
consider w ∈ Γ(γ ? (E)). According to the definition, w is parallel if ∇
∂s
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6.6
Parallel Translation
Informally, one would like to push a vector along a curve, keeping it parallel. Formally, this
is defined as:
Suppose γ : (−1, 1) → M and E is a vector bundle over M, with connection ∇. If we are
˜ ≡ 0?
given a vector w0 ∈ Tγ(0) M, can we find w ∈ Γ(γ ? (E)) such that w(0) = w0 and ∇w
The answer is yes, and we will call w the parallel translate of w0 . This is constructed as
follows.
Observe that γ ? (E) is trivial, i.e. we can pick a global frame eα ∈ Γ(γ ? (E)). Consider a
general section w ∈ Γ(γ ? (E)) as w(s) = aα (s)eα (s) and then by definition we have
˜ ∂w=∇
˜ ∂ (aα (s)eα (s))
∇
∂s
∂s
∂aα
˜ ∂ (eβ )
eα + aβ ∇
=
∂s
∂s
∂aα
=
eα + aβ Γα1,β (eα )
∂s
α
∂a
β α
=
+ a Γ1,β eα
∂s
˜ ∂ eβ ) is called a Christoffel symbol.
where Γα1,β = θα (∇
∂s
Thus w parallel means the above equation must be zero. This is a system of k first
order 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 a
neighbourhood 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)) as
before.
Let us consider the special case γ(−1) = γ(1) = p in the case of E = T R2 with ∇ being
the usual directional derivative, and then parallel translating w0 around γ would give w0 at
the end. However in general we would get back something else.
Consider for example M = S 2 and ∇ the Levi-Civita connection (to be seen soon). Here
on lines of longitude the vector points in a tangential direction and keeps the same modulus
but changes direction to remain tangential. On equatorial lines it keeps the same direction
and modulus. Thus if we prescribe a path from the north pole to the equator, then go round
a quarter of the equator and then back to the north pole on a line of longitude, we have
rotated the vector by π/2 In this case, the amount the vector is rotated equals the integral of
the Gauss curvature K in Ω, the enclosed region. However K ≡ 1 on S 2 and so the amount
of 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 holonomy
is measuring curvature.
7
7.1
Riemannian Metrics
Bundle metrics
From Q1.13 in the exercises, on a vector space V , bilinear forms B : V × V → R correspond
to elements of B ∈ V ? ⊗ V ? .
A bundle metric on a vector bundle E is an assignment to each fibre Ep of an inner
product, 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 of
Sym2 (E ? ) ⊂ E ? ⊗ E ?
Given v, w ∈ Ep we write their inner product as either hv, wih 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 on
?
E ⊗ 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(∇X v, w) + h(v, ∇X w) for all X ∈ Γ(T M)
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 a
special connection; the Levi-Civita connection and curvature.
Definition 7.3 A Riemannian metric g on a manifold M is a bundle metric on T M.
A Riemannian manifold (M, g) is a manifold with Riemannian metric. For example the
standard 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 does
is 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 there
is 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 metric
on 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 last
time 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 manifold M, 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 ∧ ... ∧ θn
corresponds to the orientation, then F(θ1 ∧ ... ∧ θk ) = θk+1 ∧ ... ∧ θn
Definition 7.7 Given an oriented Riemannian manifold, we define the (Riemannian) volume 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 manifoldR M and f : M → R a smooth map
with compact support, the integral of f is defined to be M f dV .
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 a
Riemannian manifold. It is the closest we can get to “directional derivative”.
Suppose we have a connection ∇ on T M . This extends to a connection ∇ on T ? M . given
a 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 and
so in all is Alt(∇ω)
The first requirement of the Levi-Civita connection is “torsion free” which means 1 and 2
agree (up to normalisation). Here dω = 2Alt(∇ω)
The second requirement of the Levi-Civita connection is that it is compatible with the
Riemannian metric.
Theorem 8.1 (Fundamental Theorem of Riemannian Geometry) Given a Riemannian manifold (M, g) there exists a unique connection ∇ on T M (the Levi Civita connection)
which is both torsion free and compatible with the metric. It is determined by the Koszal formula:
1
h∇X Y, Zi = [XhY, Zi − ZhX, Y i + Y hZ, Xi + hZ, [X, Y ]i + hY, [Z, X]i + hX, [Z, Y ]i]
2
We first introduce torsion properly. Take ω ∈ Γ(T ? M) and X, Y ∈ Γ(T M). Then
2Alt(∇ω)(X, Y ) = ∇ω(X, Y ) − ∇ω(Y, X)
= ∇X ω(Y ) − ∇Y ω(X)
= Xω(Y ) − ω(∇X Y ) − Y ω(X) + ω(∇Y X)
= Xω(Y ) − Y ω(X) − ω([X, Y ]) − ω(∇X Y − ∇Y X − [X, Y ])
From exercise sheet 3, dω(X, Y ) = Xω(Y ) − Y (ω(X)) − ω([X, Y ]), and comparing these
motivates us to define the torsion as follows:
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Definition 8.2 We define the torsion as
τ (X, Y ) = ∇X Y − ∇Y X − [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(
and is even in Γ(Λ2 T ? M ⊗ T M).
There is also an alternative viewpoint. We could instead define
N2
T ? M ⊗ T M)
τ := d − Alt ◦ ∇ : Γ(T ? M) → Γ(Λ2 T ? M)
but this is equivalent to before because it is also C ∞ (M) linear and is an element of Γ(T M⊗
Λ2 T ? M).
Proof (of theorem 8.1) First, assume ∇ has the required properties. Then we get from
compatibility that
h∇X Y, Zi = XhY, Zi − hY, ∇X Zi
(8.1)
and then we use the torsion free condition to write
h∇X Y, Zi = XhY, Zi − hY, [X, Z]i − hY, ∇Z Xi
(8.2)
and if we cycle X, Y, Z we get the following two equations:
h∇Z X, Y i = ZhX, Y i − hX, [Z, Y ]i − hX, ∇Y Zi
(8.3)
h∇Y Z, Xi = Y hZ, Xi − hZ, [Y, X]i − hZ, ∇X Y i
(8.4)
Now using (8.4) in (8.3) and the resulting expression in (8.2) we get
h∇X Y, Zi = XhY, Zi−hY, [X, Z]i−[ZhX, Y i−hX, [Z, Y ]i−[Y hZ, Xi−hZ, [Y, X]i−hZ, ∇X Y i]]
and simplifying gives the required equation. Thus if ∇ is a connection with the required
properties, 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 properties.
Q.E.D.
Remark Torsion only makes sense on T M (or T ? M) but compatibility of a connection with
a metric makes sense on any vector bundle.
Remark The Levi-Civita connection corresponds to directional derivatives on Rn . Take {ei }
as the standard basis. Then the connection would be determined by ∇ei ej or by h∇ei ej , ek i
so just need to check that hDei ej , ek i = 0. Remember that [ei , ej ] = 0 for all i and j.
8.2
Levi-Civita connection and Traces
Just
with a general connection, the L-C connection can be extended to
Nq like
?
T M. One of the properties was
Np
TM ⊗
∇X (trT ) = tr(∇X T )
where T is a (p, q) tensor field and tr is a contraction over any pair of T M and T ? M entries.
But now , with the metric, we can contract over any pair of indices. Thus if T = T ij ei ⊗ ej
then trT = gij T ij = tr13 tr24 g ⊗ T where tr13 means the contraction of the first and third
entries of the (2, 2) tensor g ⊗ T . Thus if ∇ is the L-C connection then
∇X (trT ) = ∇X (tr13 tr24 g ⊗ T )
= tr13 tr24 ∇X (g ⊗ T )
= tr13 tr24 [(∇X g) ⊗ T + g ⊗ (∇X T )]
= tr13 tr24 (g ⊗ ∇X T )
= tr(∇X T )
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and so the L-C connection commutes with any traces
Let u : M → Rn be an immersion. Define g = u? (gRN ) and so u is an isometric immersion.
n
We write ∇ for the L-C connection of M. We want to understand ∇X Y in terms of ∇R .
˜ be the pullback connection of ∇Rn
First, consider u? (Y ), the section of u? (T Rn ). Let ∇
under u. Then we claim that
˜ X u? (Y )]T
u? (∇X Y ) = [∇
where
T
is projection onto the tangent space. Equivalently
˜ X u? (Y ), u? (Z)iRn
g(∇X Y, Z) = h∇
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
∇
n
T
∇X Y = [∇R
X Y]
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 ∇ and M also has a Riemannian
metric g (which induces the L-C connection). Then we differentiate sections of any tensor
product of T M, T ? M, E, E ? via the product rule. For example it extends to T M ⊗ E by
E
∇X (Y ⊗ V ) = (∇LC
X Y ) ⊗ V + Y ⊗ (∇X V )
To take two derivatives, if V ∈ Γ(E) then ∇V ∈ Γ(T ? M ⊗ E) and then we can now
differentiate again
∇(∇V ) =: ∇2 V ∈ 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 ∇2 V
giving us a section of E.
Definition 8.3 The connection Laplacian (or rough Laplacian) of V ∈ Γ(E) is defined
to be
∆V := tr12 ∇2 V
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 is
the
operator. With
respect to local coordinates xi (this gives a frame
∂ Laplace-Beltrami
∂
∂
), and writing gij = g ∂xi , ∂xj , and g ij = (dxi , dxj ) then
∂xi
∆LB f := g ij
8.4
∂2f
∂f
− g ij Γkij k
i
j
∂x ∂x
∂x
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 on M and that {ei } is orthogonal. Normally upper and lower indices are
used to keep track of how objects transform as we change basis. If A transforms objects with
upper indices then A−T transforms lower indices. However if we restrict to an orthonormal
basis then we can restrict to A ∈ SO(n), i.e. A = A−T . The upshot is that we only need
consider 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(T M ⊗ T ? M) then we get
∇T ∈ Γ(T ? M ⊗ T M ⊗ T ? M) and we could write this as
∇i Tjk θi ⊗ ej ⊗ θk
This defines ∇i Tjk as coefficients. Here we are taking the tensor, applying ∇ and then
expressing it in terms of the basis, not the other way around.
If we are taking a second derivative, then we often just write
∇i ∇j Tkl
we could write the rough Laplacian of T as ∇i ∇i Tjk . However, be careful as writing ∇i ∇j
is not the same as writing ∇ei ∇ej . Think about the following:
∇i ∇j f := ∇2 f (ei , ej )
∇ei (∇ej f ) = ∇ei (ej (f )) = ∇ei (df (ej )) = (∇ei df )(ej ) + df (∇ei ej ) = ∇2 f (ei , ej ) + df (∇ei ej )
and so in the latter we have an extra term.
8.5
Divergence
N
N
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 = −∇i Tij .
In the special case that δ is applied to differential forms, we can show that, for n = dim M,
δ = (−1)nq+1 FdF
(8.5)
This gives a nice expression for the Laplace-Beltrami operator:
∆LB f = tr∇2 f = tr∇df = −δdf = FdFdf
(8.6)
A consequence of this is the Divergence theorem.
Theorem 8.5 (Divergence) Suppose M is a compact oriented manifold, and ω ∈ Γ(T ? M).
Then
Z
(δω)dV = 0
M
Proof
Z
Z
[(−1)nq+1 FdFω]F1
M
Z
= (−1)nq+1
F[FdFω]
M
Z
Stokes
=±
d(Fω) = 0
(δω)dV =
M
M
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(
N
Γ( q+1 T ? M). Then
Z
Z
hδT, Si =
hT, ∇Si
M
Nq
T ? M) and T ∈
M
Proof Define the 1-form ωk := Tk,i1 ,...,iq Si1 ,...,iq and then δω = hδT, Si − hT, ∇Si and then
integrate using the divergence theorem.
Q.E.D.
It is best to do the calculation in the proof in index notation. We restrict to the case
q = 1 for simplicity. Then ωk = Tki Si and then
δω : = −∇k ωk
= −∇k (Tki Si )
= −(∇k Tki )Si − Tki ∇k Si
= hδT, Si − hT, ∇Si
Here we use the product rule, the fact that ∇ commutes with traces etc. Without index
notation, this is a nightmare:
∇X ω = ∇X (tr23 T ⊗ S) = tr23 ∇X (T ⊗ S) = tr23 [∇X T ⊗ S + T ⊗ ∇X S]
and so
∇ω = tr34 [∇T ⊗ S] + tr24 [T ⊗ ∇S]
and so
δω = −tr∇ω = tr12 tr34 [∇T ⊗ S] + tr13 tr24 [T ⊗ ∇S]
which is what we want.
Clearly index notation is better. You could also write the integration by parts formula
in index notation:
Z
Z
(∇k Tki )Si dV =
Tki ∇k Si dV
−
M
M
This formula is maybe more explanative and looks similar to the IBP we knew from second
year.
9
9.1
Geodesics, lengths of paths, Riemannian Distance
Definition of Geodesic
Consider a map γ from an interval I ⊂ R with standard coordinate s, to a Riemannian
manifold (M, g). Informally γ will be a geodesic if it has zero acceleration.
∂
Definition 9.1 We define the velocity as γ 0 (s) := γ? ∂s
˜ for the pull-back of the L-C connection ∇ under γ. We adopt the shorthand
We write ∇
˜
Ds = ∇ ∂
∂s
˜ ∂ γ?
Definition 9.2 γ is called a geodesic if Ds (γ 0 (s)) ≡ 0, or also written ∇
∂s
∂
∂s
=0
The intuition, as I see it here, is that the velocity is the speed in R pushed forward to lie
on the manifold, and so for each point in R we have speed as a vector in the tangent space
of the manifold.
In other words, γ 0 (s) is a parallel section of γ ? (T M). In the literature it is common to
fudge notation and write X = γ 0 (s) and the equation of the geodesic as ∇X X = 0.
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Theorem 9.3 Given p ∈ M and X ∈ Tp M then there exists an open interval I ⊂ R
containing 0 and a geodesic γ : I → M with γ(0) = p and γ 0 (0) = X. Any other geodesic
˜
γ̃ : I˜ → M with γ̃(0) = p and γ̃ 0 (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 ODE
and then using existence of ODEs.
Proof Pick local coordinates {xi } near p ∈ M and without loss of generality p corresponds
to xi = 0 for all i. Write γ(s) = (x1 (s), ..., xn (s)). We will write the equation for γ being a
geodesic as an ODE for x1 , ..., xn . We can write γ 0 (s) = (ẋ1 (s), ..., ẋn (s)) or more precisely
∂
∂
as γ 0 (s) = ẋi (s) ∂x
i (γ(s)). Beware, because ∂xi (γ(s)) is not differentiating γ(s). It is the
∂
tangent vector ∂x
i at γ(s) ∈ M. By definition of the pull-back connection we have
˜ ∂ (ẋi (s) ∂ )
Ds γ 0 (s) = ∇
∂s
∂xi
∂ i
∂
˜ ∂ ∂
=
ẋ (s) i + ẋi (s)∇
∂s ∂xi
∂s
∂x
∂
∂
= ẍi (s) i + ẋi (s)∇γ 0 (s) i
∂x
∂x
∂
∂
= ẍk (s) k + ẋi (s)ẋj (s)∇ ∂
i
j
∂x
∂x ∂x
∂
= (ẍk (s) + ẋi (s)ẋj (s)Γkij ) k
∂x
=0
and so we have a nice ODE, and so by ODE theory we have a local solution if we specify
xi (0) and ẋi (0), as we have.
We now prove uniqueness. Suppose the two geodesics are not equal. They must be equal
on a neighbourhood of s = 0 by ODE theory. Thus there must be a value s0 > 0 where the
geodesics diverge. At this point, both geodesics have the same velocity, and so by applying
the 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 exists
a well defined maximal geodesic associated with an arbitrary p ∈ M and X ∈ Tp M
(associated with an arbitrary X ∈ T M for which we define π(X) = p) i.e. it cannot be
extended to any larger interval.
Definition 9.4 The Riemannian manifold (M, g) is geodesically complete if the maximal 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) we
will see that geodesic completeness is equivalent to completeness with respect to a natural
metric 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 T M
to M which takes X ∈ T M and maps it to γ(1) where γ is the unique geodesic associated
with X.
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Definition 9.5 We define DM to be the subset of T M consisting of all X ∈ T M for which
there 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 the
unique geodesic associated with X.
We define DM in order to make the exponential map make sense. For example, if
M = S 2 then DM = T M.
Remark If we restrict to one fibre Tp M ∩ DM then we write the exponential map as
expp : Tp M ∩ DM → M
On the sphere expp maps D(0, π), the ball of radius π > 0 in the tangent space in Tp M,
onto S 2 \ {south pole}.
Lemma 9.6 (Properties of the exponential map.) The following properties hold.
1. DM is an open subset of T M containing all zero length vectors, and each fibre is “star
shaped”, i.e. each point can be connected to 0 ∈ Tp M by a straight line.
2. Given X ∈ DM, the curve s 7→ exp(sX) is a geodesic defined at least for s ∈ [0, 1] and
is 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 (p) > 0 such
that expp is a diffeomorphism from D(0, ) ⊂ Tp M to its image in M.
We will call the supremum of all such the injectivity radius at p. Note that this can be
infinity. For example, on the sphere this is π. To prove this lemma we will need the inverse
function theorem for manifolds.
Proof (sketch) We want to take the derivative of expp : Tp M ∩ DM → M at 0 ∈ Tp M.
We get that this is a map
d(expp ) : T0 (Tp M ∩ DM) → Tp M
but there is a natural identification of T0 (Tp M) with Tp M and so we can write
d(expp ) : Tp M → Tp M
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 d d(expp ) = expp ◦τ (t) = expp tX = γX (t)|t=0 = X
dt t=0
dt t=0
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)) a
geodesic 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 Tp M.
Define a chart φ by
φ−1 (x) = expp (xi ei )
where x = (x1 , ..., xn ) are the coordinates of xi ei .
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 ∇
∂
p. Also, gij = δij and so in this case { ∂x
i } are orthogonal.
9.4.1
∂
∂xi
= 0 at
Uniform Normal Neighbourhoods
It is a problem in lemma 9.7 that the injectivity radius could depend horribly on p; it could
be discontinuous with respect to p.
Lemma 9.9 (Uniform normal neighbourhood lemma) For all p ∈ M and any neighbourhood U of p there exists a neighbourhood W ⊂ U of p such that there exists a δ > 0 such
that for all q ∈ W , the restriction of expq to the ball D(0, δ) ⊂ Tq M is a diffeomorphism
and expq (D(0, δ)) ⊃ W .
Proof [Non-examinable?] The key idea here is instead of linearising expp at 0 ∈ Tp M we
linearise X 7→ (π(X), exp(X)) at 0 ∈ Tp M within T M. 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 local
frames. 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,
∇LC ei = 0 for all i.
Proof (sketch) Pick any ONB {ei } of Tp M. Extend {ei } to a neighbourhood of p (any
geodesic ball) by parallel translating each ei radially along geodesics. By Q3.10, this gives
an 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
Z
L(γ) =
b
|γ 0 (s)|ds
a
1
where |X| = g(X, X) 2
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] → M
such that there exist finite subdivision of [a, b] where a < a0 < a1 < ... < ak < b such that
γ|[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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MA4C0 Differential Geometry Lecture Notes Autumn 2012
9.7
Riemannian distance
Assume that (M, g) is connected. Given p, q ∈ M, define Σp,q to be the set of all admissible
curves 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) = inf (L(γ))
γ∈Σp,q
Observe that this definition implicitly contains the Riemannian metric g. We use Q5.7 to
check 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 metric
space, 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(γ̃) where
γ̃ 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 have
constant speed. Note that we can always parametrise a curve to have constant speed without
changing length.
Theorem 9.15 If an admissible curve is minimising and it has constant speed then it is a
geodesic. 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 a
smooth 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
d
The goal is to find a formula for dη
L(F (η, ·))|η=0 . We will assume that γ = F (0, ·) has unit
0
speed parametrisation, i.e. |γ (s)| = 1 for all s ∈ [a, b]. We will use the notation
∂F
∂F
∂
=
(η, s) = F?
X=
∂s
∂s
∂s
and that
∂F
∂F
Y =
=
(η, s) = F?
∂η
∂η
∂
∂η
Observe that |X| = 1 when η = 0 and X is the tangent vector along the curves and Y is the
variation vector field.
Theorem 9.16 Given a smooth F as above, for which F (0, ·) has unit speed parametrisation,
d
L(F (η, ·))|η=0 = [hX, Y i]s=b
s=a −
dη
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Z
b
hY, Ds Xids
a
MA4C0 Differential Geometry Lecture Notes Autumn 2012
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
Ds Y = Dη X
∂F
or equivalently that Ds ∂F
∂η = Dη ∂s , i.e. in this case the derivatives commute.
Proof By the definition of L, we have
b
Z
Z
|X|ds =
L(F (η, ·)) =
b
1
g(X, X) 2 ds
a
a
and so using the fact that |X| = 1 we get that
b
1
− 12 ∂
g(X, X)
g(X, X)ds
∂η
a 2
η=0
Z b
=
g(Dη X, X)ds
d
L(F (η, ·))|η=0 =
dη
Z
a
η=0
b
g(Ds Y, X)ds
=
a
η=0
Z b
∂
[g(X, Y )] − g(Y, Ds X) ds
=
∂s
a
Z b
= [g(X, Y )]s=b
−
g(Y, Ds X)ds
s=a
a
Z b
= [hX, Y i]s=b
−
hY, Ds Xids
s=a
Z
a
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 then
Ds γ 0 (s) ≡ 0. Let us suppose that Y ∈ Γ(γ ? (T M)) 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
F (η, a) = γ(a), F (η, b) = γ(b) and ∂F
∂η (0, ·) = Y , and as usual we set X = ∂s .
By minimising hypothesis we have
d
L(F (η, ·))|η=0 = 0
dη
Rb
d
and the first variation formula gives dη
L(F (η, ·))|η=0 = − a hY, Ds Xids due to the end
Rb
points not changing. Therefore a hY, Ds Xids = 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. Set
Y (s) = φ(s)Ds X and then
Z
b
Z
φ(s)hDs X, Ds Xids =
a
b
φ(s)|Ds X|2 ds = 0
a
and so Ds X ≡ 0 i.e. γ is a geodesic since Ds γ 0 (s) = 0 as X(0, s) =
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∂F
∂s (0, s)
= γ 0 (s).
MA4C0 Differential Geometry Lecture Notes Autumn 2012
9.9
Geodesics are locally minimising
Definition 9.17 A curve γ : I → M from some interval I ⊂ R is called locally minimising if for all c ∈ I there exists a neighbourhood U ⊂ I of c such that for all a0 , b0 ∈ U then
the restriction of γ|[a0 ,b0 ] 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 picking
normal coordinates {xi } centred at p and setting
r(x) =
n
X
!1
2
i 2
(x )
i=1
This is well defined because it is independent of the choice of orthonormal basis {ei } from
the 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 (sxi ei ). Then γ 0 (0) = xi ei and therefore |γ 0 (s)| = |γ 0 (0)| = |xi ei | = r(x) and therefore
L(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 these
other 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 0 (s) = xi ∂x
i |γ(s) which has norm r. We thus define a
unit radial vector field to be
∂
xi ∂
=
∂r
r ∂xi
We could have tried to define this as ∇r = (dr)# , where sharp is the natural isomorphism
∂
from T ? M to T M. 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 ∈ Tq M q 6= p we need to show
∂
∂
dr(X) = h∇r, Xi = h ∂r
, Xi. If we set X = ∂r
then we have the right hand side equal to
RHS =
and
xi
LHS = dxi
r
∂ ∂
,
∂r ∂r
xj ∂
r ∂xj
2
∂ = = 1
∂r
=
xi xj
δij = 1
r
and so they are equal. We can decompose a general X as a radial vector and a vector tangent
to the geodesic sphere at p through q. Therefore it remains to prove the above for X a tangent
vector to the geodesic sphere through q, i.e. for X such that h∇r, Xi = dr(X) = X(r) = 0.
Thus we must prove that
∂
h , Xi = 0
∂r
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MA4C0 Differential Geometry Lecture Notes Autumn 2012
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 write
q = expp (V ) for V ∈ ∂D(0, R).
Let X ∈ Tq M be tangent to the geodesic sphere ∂D(0, R) i.e. X(R) = 0. expp is
a diffeomorphism on geodesic balls and therefore there exits a W ∈ TV (Tp M) such that
d 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 and
σ 0 (0) = W . Consider F : (−δ, δ) × [0, 1] → M defined by F (η, s) = expp (sσ(η)). Therefore
for all η ∈ (−δ, δ) then F (η, ·) is a radial geodesic with speed R. The objective is to show
that 1r ∂F
∂s (0, 1) is orthogonal to X.
We proceed with a few computations:
∂F
d
(0, 0) =
|η=0 expp (0) = 0
∂η
dη
∂F
d
(0, 1) =
|η=0 expp (σ(η)) = X
∂η
dη
by construction. Then
∂
∂s
∂F ∂F
,
∂η ∂s
=
∂F ∂F
Ds
,
∂η ∂s
+
∂F
∂F
, Ds
∂η
∂s
but since F (η, ·) is a geodesic, we have Ds ∂F
∂s = 0 and the symmetry lemma (Q5.9) gives
∂F
=
D
and
thus
we
have
Ds ∂F
η ∂s
∂η
∂
∂s
∂F ∂F
,
∂η ∂s
=
∂F ∂F
Dη
,
∂s ∂s
1 ∂
=
2 ∂η
∂F ∂F
,
∂s ∂s
1 ∂
=
2 ∂η
∂F 2
∂s = 0
since F (η, ·) is a geodesic so ∂F
∂s is constant, so its rate of change is zero.
∂F
At s = 0 we have ∂η (0, 0) = 0 and therefore
∂F ∂F ,
=0
∂η ∂s s=0,η=0
and therefore if we integrate from s = 0 to s = 1 to get, for η = 0,
Z 1
∂
∂F ∂F
∂F ∂F ∂F ∂F ,
ds =
,
−
,
=0
∂η ∂s
∂η ∂s s=1,η=0
∂η ∂s s=0,η=0
0 ∂s
and so hX, ∂F
∂s i = 0.
9.9.2
Q.E.D.
Radial Geodesics are Initially minimising
Lemma 9.20 Suppose p ∈ M and ∈ (0, ∞) is no more than the injectivity radius. Then for
all q ∈ expp (D(0, )) (i.e. q = expp (X) for X ∈ D(0, )), 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 piecewise
smooth curves from p to q modulo reparametrisation.
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MA4C0 Differential Geometry Lecture Notes Autumn 2012
Proof Define R := |X| and consider the radial geodesic with arclength parametrisation
0
γ : s 7→ expp ( sX
R ). 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 being
the length of γ.
We can assume the other curve is smooth (easy to extend to piecewise smooth). Also, by
chopping off the start of the curve we may assume it never returns to p (this only decreases
the 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 σ
has length at least R.
∂
We decompose orthogonally to get σ 0 (s) = a(s) ∂r
+ V (s) for a(s) ∈ R and V (s) ⊥ ∇r
and this is for s > 0. Therefore
|σ 0 (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) = hσ 0 (s), ∇ri = dr(σ 0 (s))
and so
L(σ) = lim L(σ|[,1] )
→0
Z 1
|σ 0 (s)|ds
= lim
→0 Z 1
dr(σ 0 (s))ds
≥ lim
→0 Z 1
d
= lim
[r(σ(s))]ds
→0 ds
= lim [r(σ(1)) − r(σ())]
→0
= R − r(σ(0))
=R
It thus remains to show uniqueness. Suppose we have some other curve from p to q
with the least possible length. By the beginning of the proof, our new “competitor” curve
∂
has that it doesnt return to p, and never leaves expp (D(0, )) and that σ 0 (s) = ∂r
(if we
0
reparameterise 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 Riemannian distance function d(x, p).
Thus geodesic balls expp (D(0, r)) can be written B(p, r) where more generally we define
B(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 find
a neighbourhood about c where γ is minimising. Without loss of generality we can assume
that I is an open interval.
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MA4C0 Differential Geometry Lecture Notes Autumn 2012
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 that
contains the whole of W .
Let (a, b) ⊂ I be the connected component of the preimage of W , i.e (γ −1 (W )) containing
c. We claim that if a < a0 < b0 < b then γ|[a0 ,b0 ] is minimising, with c ∈ (a0 , b0 ).
To show this note that γ([a0 , b0 ]) is contained in W by construction and therefore within a
geodesic ball of radius δ centred at γ(a0 ). By construction γ|[a0 ,b0 ] is a radial geodesic starting
at γ(a0 ), remaining within the geodesic ball centred at γ(a0 ). By lemma 9.20, radial geodesics
are 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 metric
space 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 minimising
geodesic. 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 by
defining the curvature of a connection and then we will be able to take the curvature of the
Levi-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(T M) × Γ(T M) × Γ(E) → Γ(E) written (X, Y, v) 7→ R(X, Y )v defined by
R(X, Y )v := −∇X ∇Y v + ∇Y ∇X v + ∇[X,Y ] v
It should be noted that half of everybody takes the opposite sign convention to that
used above. Q6.1 says curvature is not just a map of this form. It is also a tensor. In fact
R(X, Y ) at p ∈ M only depends on X, Y, v at p and is not dependent on values in some
neighbourhood of p. Thus it is in fact a section of T ? M ⊗ T ? M ⊗ E ? ⊗ E. Also notice the
symmetry
R(X, Y )v = −R(Y, X)v
and so R is in fact a section of Λ2 T ? M ⊗ End(E). We can also view R as the endomorphism
of 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 it
around 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 curvature
R is a section of T ? M ⊗ T ? M ⊗ T ? M ⊗ T M i.e. a (1, 3) tensor field. In this context we can
also use the metric (via musical isomorphism ) to define the (0, 4) tensor Rm (X, Y, W, Z) =
hR(X, Y )W, Zi. Sometimes we write Rm (g) if we want to emphasise which metric we are
taking the curvature of.
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MA4C0 Differential Geometry Lecture Notes Autumn 2012
Remark Rm is invariant under isometries, i.e. if φ is an isometry then Rm (φ? g) = φ? (Rm (g)).
All the Riemannian metrics we are looking at are invariant under isometries (by construction).
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 )hW, Zi
= R(X, Y )(trW ⊗ Z)
= trR(X, Y )(W ⊗ Z)
= tr[(R(X, Y )W ) ⊗ Z + W ⊗ (R(X, Y )Z)]
= hR(X, Y )W, Zi + hW, R(X, Y )Zi
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 2
columns 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 ∇V Rm (W, X, Y, Z) + ∇V Rm (V, W, Y, Z) + ∇W Rm (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 ⊂ Tp M. Take an orthonormal frame X, Y for π. Then the sectional curvature of M at p with respect to π
is
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 have
positive sectional curvature.
If M is a two dimensional manifold then we have to take π = Tp M and K is then the
Gauss curvature at p.
Given K(π) for all p ∈ M and π ⊂ Tp M 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, and
so we want nicer measures of curvature which still contain enough information.
Definition 10.5 The Ricci curvature Ric : Γ(T M) × Γ(T M) → C ∞ (M) is the section
of sym2 (T ? M) defined by
Ric(X, Y ) = tr(Rm (X, ·, Y, ·))
P
In other words Ric(X, Y ) = tr{Z 7→ R(X, Z)Y } = i Rm (X, ei , Y, ei ) for {ei } an orthonormal frame field.
Definition 10.6 The scalar curvature is defined to be R ∈ C ∞ (M) by
R = trg (Ric)
In other words R = g ij Rij where Rij dxi ⊗ 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 =
− 23 Rij
In two dimensions, the scalar curvature R is twice the Gauss curvature.
We consider the second Bianchi identity again:
∇V Rm (W, X, Y, Z) + ∇V Rm (V, W, Y, Z) + ∇W Rm (X, V, Y, Z) = 0
If we trace over W and Y and then over X and Z we get
∇V R + 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 for
λ : M → R, i.e. by taking trace of this giving R = λn. Then we have
R
Ric = g
n
Lemma 10.8 (Schur) The two above definitions of an Einstein metric are the same, so
long as dim M =
6 2
Proof We take the divergence to get
R
1
1
dR
δRic = δ
g = δ(Rg) = − tr(∇Rg) = −
n
n
n
n
Then using lemma 10.7
n
dR
2
and if n = 2 this is vacuous. If n 6= 2 then dR = 0 and so R is constant
dR = −nδRic =
10.6
Q.E.D.
Curvature Operator
Given the symmetries of Rm (1 and 2 in lemma 10.2) we can define a linear map
R : Λ2 (T M) → Λ2 (T M)
called the curvature operator, defined by
hR(X ∧ Y ), W ∧ Zi = Rm (X, Y, W, Z)
By symmetry 4 of lemma 10.2 we have that R is symmetric. Therefore we can diagonalise
R. 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 condition 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 S 2 or RP 2 .
Theorem 10.9 (Böhn-Wilking 2006) (non exam) A simply connected compact manifold
with positive curvature operator must be diffeomorphic to S n .
10.7
Curvature in Index notation
We introduce several conventions for the coefficients of curvature with respect to coordinates:
Rm = Rijkl dxi ⊗ dxj ⊗ dxk ⊗ dxl
Ric = Rij dxi ⊗ dxj
and therefore Rij = g kl Rikjl and R = g ij Rij = g ij g kl Rikjl .
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 redo
the contracted second Bianchi. This is
∇i Rjklm + ∇k Rijlm + ∇j Rkilm = 0
and tracing over j and l and k and m gives
∇i R − ∇k Rik − ∇j Rij = 0
which is what we want in index notation.
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11
11.1
Conquering Calculation
Switching Derivations
Given some general tensor T , if we apply ∇ twice to get ∇2 T ∈ Γ(T ? M⊗T ? M⊗...) and then
apply first T ? M ⊗ T ? M part to X, Y ∈ Γ(T M) we write the result ∇2X,Y T . For example if
T is a 1-form then ∇2X,Y T := ∇2 T (X, Y )
∇2X,Y T = (∇X ∇T )(Y ) − ∇X (∇T (Y )) − ∇T (∇X Y ) = ∇X ∇Y T − ∇∇X Y T
and as a consequence
−∇2X,Y + ∇2Y,X = −(∇X ∇Y − ∇∇X Y ) + (∇Y ∇X − ∇∇Y X )
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 a
vector 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 + ∇2Y,X w)(Z) = −w[R(X, Y )(Z)]
i.e. the way we switch derivatives of a 1-form. More generally, for any A ∈ Γ(⊗k T ? M) we
have
(−∇2X,Y A + ∇2Y,X A)(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 ∇2X,Y w = ∇2 w(X, Y ) we have this equal to
∇2X,Y w = (∇i ∇j wk )X i Y j ek
and also
R(X, Y )w = −∇i ∇j wk X i Y j ek + ∇j ∇i wk X i Y j ek
Also R(X, Y )w = Rijkl X i Y j wk el and we would normally then write that
Rijkl wk = −∇i ∇j wk + ∇j ∇i wk
or
∇i ∇j wk = ∇j ∇i wk + Rijkl wl
More generally we have
∇i ∇j Ak1 ...km = ∇j ∇i Ak1 ...km + Rijkl Alk2 ...km + Rijkl Ak1 lk3 ...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 stuff
about the topology.
Proposition 11.1 Suppose f : M → R with M a Riemannian manifold. Then
1
∆|df |2 = |Hess(f )|2 + hdf, d|∆LB f |i + Ric(∇f, ∇f )
2
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 hdf, dhi would be h∇f, ∇hi
3. In index notation hdf, dhi = (∇i f )(∇i h)
4. In index notation ∆LB f = ∇i ∇i f
5. In index notation |Hess(f )|2 = (∇i ∇j f )(∇i ∇j f )
6. Hess is symmetric i.e. ∇i ∇j f = ∇j ∇i f
7. Ricci Identity ∇i ∇i wk = ∇j ∇i wk + Rijkl wl
Proof
1
1
∆|df |2 = ∇i ∇j ((∇j f )(∇j f ))
2
2
= ∇i [(∇i ∇j f )(∇j f )]
= (∇i (∇i ∇j f ))(∇j f ) + (∇i ∇j f )(∇i ∇j f )
= (∇i ∇j ∇i f )(∇j f ) + |Hess(f )|2
= (∇j (∆LB f ))(∇j f ) + Rjl (∇l f )(∇j f ) + |Hess(f )|2
= hd|∆LB f |, df i + 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 the
following basic set up. Let γ : [a, b] → M be a smooth curve, and F : (−δ, δ) × [a, b] → M
is the variation and consider the length L(F (η, ·)). Previously we assumed that γ had unit
speed. Now we will assume that it is a geodesic. Also assume that the end points are kept
fixed 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 − hY, γ 0 (s)iγ 0 (s)
in other words this is Y minus the part parallel to γ 0 .
∂F
Suppose that X = ∂F
∂s and Y = ∂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 with
fixed endpoints, we have
Z bh
i
d2 ⊥ 2
⊥ 0
⊥ 0
L(F
(η,
·))
=
|D
Y
|
−
R
(Y
,
γ
(s),
Y
,
γ
(s))
ds
s
m
dη 2 η=0
a
Note that the second derivative of length on a negatively curved manifold is positive, and so
all 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 for
first variation formula except we cannot yet set η = 0. Recall that, from the definition,
Z b
Z b
∂F 1
ds =
g(X, X) 2 ds
L(F (η, ·)) =
∂s a
a
and also observe that
Thus
d
L(F (η, ·)) =
dη
d
dη [g(X, X)]
b
Z
a
= 2g(Dη X, X) = 2g(Ds Y, X) by the symmetry lemma.
1 d
1
g(X, X)− 2 (g(X, X))ds =
2
dη
Z
b
1
g(X, X)− 2 g(Ds Y, X)ds
a
Differentiate with respect to η again and now set η = 0 to get
Z b
1
1
d2 d
− 32
L(F (η, ·)) =
− g(X, X)
g(X, X) g(Ds Y, X) + g(X, X)− 2 ×
2
dη η=0
2
dη
a
× (g(Dη Ds Y, X) + g(Ds Y, Dη X))ds
Z
b
−
=
a
Z
1 d
g(X, X) g(Ds Y, X) + g(Dη Ds Y, X) + g(Ds Y, Dη X)ds
2 dη
b
−[g(Ds Y, X)]2 + g(Dη Ds Y, X) + g(Ds Y, Ds Y )ds
=
a
Z
b
−[g(Ds Y, X)]2 + |Ds Y |2 + g(Ds Dη Y, X) + g(R̃(
=
a
Z
b
−[g(Ds Y, X)]2 + |Ds Y |2 +
=
a
∂ ∂
,
)Y, X)ds
∂s ∂η
∂
[g(Dη Y, X)]
∂s
− g(Dη Y, Ds X) + g(R̃(X, Y )Y, X)ds
Z
=
b
−[g(Ds Y, X)]2 + |Ds Y |2 + Rm (X, Y, Y, X)ds
a
since Ds X = 0 for η = 0 as we are on a geodesic, and the other term is zero as it depends
on the change of the endpoints, which is zero.
Now Y = Y ⊥ + Y T with Y T = hY, XiX and so Ds (Y T ) = hDs Y, XiX as Ds X = 0 for
η = 0. Thus
|(Ds Y )T |2 = |hDs Y, XiX|2 = hDs Y, Xi2
and also
(Ds Y )⊥ = Ds Y − hDs Y, XiX = Ds (Y ⊥ ) + Ds (Y T ) − hDs Y, XiX = Ds (Y ⊥ )
and thus
|Ds Y |2 = |(Ds Y )⊥ |2 + |(Ds Y )T |2 = |Ds Y ⊥ |2 + g(Ds Y, X)2
We then conclude that
Z b
Z b
d2
⊥ 2
|
L(F
(η,
·))
=
|D
Y
|
+
R
(X,
Y
Y,
X)ds
=
|Ds Y ⊥ |2 + Rm (Y, γ 0 (s), Y, γ 0 (s))ds
η=0
s
m
dη 2
a
a
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MA4C0 Differential Geometry Lecture Notes Autumn 2012
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 only
depends on Y at η = 0. Analogous fact in finite dimensions is this. If σ(η) is a smooth curve
in R2 with σ(0) = 0 and L : R2 → R a smooth function then
d2 L(σ(η)) = (HessL)(σ 0 (0), σ 0 (0))
dη 2 η=0
provided ∇L = 0 at 0. Otherwise we would get a σ 00 term.
A consequence of this is given a minimising geodesic γ : [a, b] → M then for any Y ∈
Γ(γ ? (T M)) with Y (a) = 0 = Y (b) and Y (s) ⊥ γ 0 (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
| L(F (η, ·))
dη 2 η=0
≥ 0 and therefore we have the following.
Corollary 12.3 Under the circumstances above
Z b
Q(Y ) :=
|Ds Y |2 − Rm (Y, γ 0 (s), Y, γ 0 (s))ds ≥ 0
a
13
Classical Riemannian Geometry Results: Bonnet-Myers
Definition 13.1 The diameter of a Riemannian connected manifold is defined to be
diam(M, g) = sup dg (p, q) ∈ (0, ∞]
p,q∈M
Theorem 13.2 (Myers) Suppose r > 0 and (M, g) is a complete connected Riemang i.e. if X ∈ T M then Ric(X, X) ≥
nian manifold of dimension n such that Ric ≥ n−1
r2
n−1
g(X,
X).
Then
M
is
compact
and
2
r
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
g.
r2
Proof Start by proving the diameter estimate. Pick arbitrary points p, q ∈ M and define
L = 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] → M
such that γ(0) = p and γ(L) = q, and with unit speed. Pick an orthonormal basis {ei } of Tp M
with en = γ 0 (0). Parallel translate this frame {ei } along γ to give sections ei ∈ Γ(γ ? (T M))
and note that for all s ∈ [0, L] we have {ei (s)} is an orthonormal basis of Tγ(s) M. Note for
all s that en (s) = γ 0 (s) because en (0) = γ 0 (0) by construction and both en (s) and γ 0 (s) are
parallel (satisfy same ODEs).
Define for all i ∈ {1, ..., n − 1}
πs ei (s)
Yi (s) = sin
L
and observe that Yi (0) = 0 = Yi (L) and that Yi (s) ⊥ γ 0 (s).
Apply corollary 12.3 to get that
Z L
Q(Yi ) =
|Ds Yi |2 − Rm (Yi , γ 0 (s), Yi , γ 0 (s))ds ≥ 0
0
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MA4C0 Differential Geometry Lecture Notes Autumn 2012
but Ds Yi =
π
L
cos
πs
L
ei (s) + sin
πs
L
×0=
|Ds Yi |2 =
π
L
Rm (Yi , γ 0 (s), Yi , γ 0 (s)) = sin2
L
Z
0≤
0
πs
L
ei (s) and therefore
π2
2 πs
cos
L2
L
and also
therefore
cos
πs L
Rm (ei , γ 0 (s), ei , γ 0 (s))
π2
2 πs
2 πs
cos
−
sin
Rm (ei , γ 0 (s), ei , γ 0 (s))ds
L2
L
L
and then sum over i = 1, ..., n − 1 and note that Rm (en , γ 0 (s), en , γ 0 (s)) = 0 and so
Z
L
0≤
0
but Ric(γ 0 (s), γ 0 (s)) ≥
π 2 (n − 1)
2 πs
2 πs
cos
−
sin
Ric(γ 0 (s), γ 0 (s))ds
L2
L
L
n−1
g(γ 0 (s), γ 0 (s))
r2
Z
0≤
0
L
=
n−1
r2
and therefore
n−1
π 2 (n − 1)
2 πs
2 πs
cos
−
sin
ds
L2
L
r2
L
and so
0≤
π 2 (n − 1) n − 1
−
L2
r2
i.e. L2 ≤ π 2 r2 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 we
have the result.
Q.E.D.
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