University of Cambridge

Mathematics Tripos

Part III Differential Geometry

Michaelmas, 2018

Lectures by A. Kovalev

Notes by Qiangru Kuang Contents

Contents

1 Manifolds 2

2 Matrix Lie groups 6

3 Tangent space to manifolds 9 3.1 Lie algebra ...... 10 3.2 Left-invariant vector field ...... 15

4 Submanifolds 18

5 Differential forms 22 5.1 Orientation of manifolds ...... 24 5.2 Exterior derivative ...... 25 5.3 Pullback of differential forms ...... 26 5.4 de Rhan cohomology ...... 27 5.5 Basic integration on manifolds ...... 29

6 Vector bundles 31 6.1 Structure group and transition functions ...... 32 6.2 Principal bundles ...... 33 6.3 Hopf bundle ...... 35 6.4 Pullback of a vector bundle or a principal bundle ...... 36 6.5 Bundle morphism ...... 37

7 Connections 39 7.1 Curvature ...... 44

8 Riemannian geometry 46 8.1 Geodesics ...... 49 8.2 Curvature of a Riemannian metrix ...... 52

1 1 Manifolds

1 Manifolds

We want to generalise curves and surfaces in R2. A curve is a map 훾 ∶ R ⊇ 퐼 → R2 or R3 that satisfies certain properties we’ll find out in a minute. Clearly continuity is necessary but not sufficent, as evidenced by the famous Peano space-filling curve. Smoothness is not quite enough either, as 푡 ↦ (푡2, 푡3) has a cusp at the origin. The correct requirement will be that 훾 has regular parame- terisation, i.e. | ̇훾(푡)| ≠ 0 for all 푡. Similarly, a surface should be defined as as a map 푟 ∶ R2 ⊇ 퐷 → R3 with a ∞ 휕푟 휕푟 regular parameterisation, i.e. 푟 ∈ 퐶 (퐷) and ∣ 휕푢 × 휕푣 ∣ ≠ 0 for all (푢, 푣) ∈ 퐷. We may follow this route and generalise to (hyper)surfaces in R푛, which will be a generalisation of classical differential geometry of curves and surfaces in R3. The good thing is that we can readily apply calculus and it is easy to construct these objects. However, it does suffer from the prolblem of different parameteri- sations give rise to different geometric objects, as well as some surface requiring more than one parameterisation. Although these can be bypassed more or less, the more serious drawback is the extra technical complexity determined by the higher dimension of the ambient space. A better concept is smooth manifolds. We first begin with a review of topo- logical structure, which every manifold possesses.

Definition (topological space). A topological space 푀 is a choice of class of the open sets such that 1. ∅ and 푀 are open,

2. if 푈 and 푈 ′ are open then so is 푈 ∩ 푈 ′, 3. for anly collection of open sets, the union is open.

In this course, we always require a topological space to be Hausdorff and second countable.

Definition (local coordinate chart). A local coordinate chart on a topolog- ical space 푀 is a homeomorphism 휑 ∶ 푈 → 푉 where 푈 ⊆ 푀 and 푉 ⊆ R푑 are open. 푈 is a coordinate neighbourhood.

Definition (퐶∞-differentiable structure). A 퐶∞-differentiable structure on a topological space 푀 is a collection of charts {휑훼 ∶ 푈훼 → 푉훼} where 푑 푉훼 ⊆ R for all 훼 such that

1. {푈훼} covers 푀, i.e. 푀 = ⋃ 푈훼, −1 ∞ 2. compatibility condition: for all 훼, 훽, 휑훽 ∘ 훽훼 is 퐶 wherever defined, i.e. on 휑훼(푈훼 ∩ 푈훽),

3. maximality: if 휑 is compatible with all the 휑훼’s then 휑 is in the collection.

The collection of charts is called an atlas. −1 Note that 2 implies that 휑훽 ∘ 휑훼 is a diffeomorphism.

2 1 Manifolds

Definition (manifold). A manifold is a Hausdorff, second countable topo- logical space with a 퐶∞-differentiable structure.

Remark. 1. In practice, we almost never specify the topological structure. Instead, we can induce a topology from a 퐶∞ structure by declaring 퐷 ⊆ 푀 open if 푑 and only if for all (휑훼, 푈훼), 휑훼(푈훼 ∩ 퐷) is open in R . 2. We may replace 퐶∞ by 퐶푘 for 푘 > 0 finite. If we set 푘 = 0, the objects be- come topological manifolds. On the other hand, use C푛 and holomorphic maps we get complex manifolds. 3. Requirement of being Hausdorff and second countable are for rather tech- nical reasons. In some cases we may drop Hausdorffness requirement, and such examples do arise naturally. However in that case we lose uniqueness of limits. Similarly non-second countable space, such as the disjoint union of uncountably many R푛, can become manifold-like structure if we relax the definition. Example.

1. R푑 covered by the single chart 휑 = id. 푆푛 = {퐱 = (푥 , … , 푥 ) ∈ 푛+1 ∶ ∑푛 푥2 = 1} 2. Unit sphere 0 푛 R 푖=0 푖 . The charts are stereographic projections 1 휑(퐱) = (푥1, … , 푥푛) (1) 1 − 푥0 1 휓(퐱) = (푥1, … , 푥푛) (2) 1 + 푥0 where 휑 is defined for all points on 푆푛 except (1, 0, … , 0) and similar for 휓. Suppose 휑(푃 ) = 푢, 휓(푃 ) = 푣, then by basic geometry 푣 = 휓 ∘ 휑−1(푢) = 푢 ‖푢‖2 .

3. Given 푀1, 푀2 manifolds of dimension of 푑1 and 푑2, 푀1 ×푀2 is a manifold 1 1 of dimension 푑1 + 푑2. We define the 푛-torus to be 푇푛 = 푆⏟⏟⏟⏟⏟× ⋯ × 푆 . 푛 4. An open subset 푈 of a manifold 푀 is a manifold. 5. The real projective space,

R푃 푛 = {all straight lines through 0 in R푛+1}.

The points in the space are 푥0 ∶ ⋯ ∶ 푥푛 where 푥푖’s are not all zero. Note that 푥0 ∶ ⋯ ∶ 푥푛 = 휆푥0 ∶ … 휆푥푛 for all 휆 ≠ 0. As per a previous remark, we shall induce the topology by ∞ a 퐶 structure. The charts are (푈푖, 휑푖) where 푈푖 = {푥푖 ≠ 0}, and

푥0 ̂ 푥푛 휑푖(푥0 ∶ ⋯ ∶ 푥푛) = ( ,…, 푖, … , ) 푥푖 푥푖

3 1 Manifolds

̂ 푛 where 푖 denotes that the 푖th coordinate is omitted. The 푈푖’s cover R푃 so we are left to check compatibility. For 푖 < 푗, 푦 1 푦 −1 1 ̂ 푛 휑푗 ∘ 휑푖 ∶ (푦1, … , 푦푛) ↦ 푦1 ∶ … , 1⏟ ∶ 푦푛 ↦ ( ,…, ,…, 푗, … , ) 푖th 푦푗 푦푗 푦푗 is smooth. Since 푖 and 푗 are arbitrary, R푝푛 is a 푛-manifold. Similarly we can check C푃 푛 is a 2푛-manifold, and H푃 푛 is a 4푛-manifold (but this is a bit tricky due to noncommutativity). 6. Grassmannians (over R or C): we define Gr(푘, 푛) to be all 푘-dimensional subspaces of the 푛-dimensional vector space, which generalises the projec- tive space. For real vector spaces for example, R푃 푛 = Gr(1, 푛 + 1). We can check Gr(푘, 푛) is a manifold of dimension 푘(푛 − 푘). The construction is a bit technical so we will give example of one chart. Let 푈 be the 푘-subspaces obtainable as the span of rows of 푘×푛 matrices of the form (퐼푘 ∗), and the local coordinate maps a basis of such 푘-subspace to the 푘 × (푛 − 푘) block ∗. Since the frist 푘 rows are linearly independent, we call 푈 = 푈1<2<⋯<푘. More generally the domain of charts take the 푈 퐶∞ form 1≤푖1<⋯<푖푘≤푛. It is an exercise to check that this gives a valid structure. 푦 7. A non-example: define an equivalence relation (푥, 푦) ∼ (휆푥, 휆 ) for all 휆 ≠ 0 and define 푋 ∶= R2/ ∼. {푥푦 = 푐} is one equivalence class if 푐 ≠ 0. If 푐 = 0, {푥푦 = 0} is three classes {(푥, 0) ∶ 푥 ≠ 0}, {(0, 0)}, {(0, 푦) ∶ 푦 ≠ 0}. Thus 푋 ≅ (−∞, 0) ∪ {0′, 0″, 0‴} ∪ (0, ∞). Define charts (푖) 휑푖 ∶ (−∞, 0) ∪ 0 ∪ (0, ∞) → R in the obvious ways. We can check that it gives 푋 a valid 퐶∞ structure, except that the induced topology is non-Hausdorff! 8. For a non-second countable example, see example sheet 1 Q12.

Definition (smooth map). Let 푀, 푁 be manifolds, a continuous map 푓 ∶ 푀 → 푁 is smooth or 퐶∞ if for any 푝 ∈ 푀, there exists charts (푈, 휑) on 푀, (푉 , 휓) on 푁 such that 푝 ∈ 푈, 푓(푝) ∈ 푉 and

휓 ∘ 푓 ∘ 휑−1

is smooth wherever it is defined, i.e. on 휑(푈 ∩ 푓−1(푉 )) ⊆ R푛 where 푛 = dim 푀. Remark. Note that by continuity, 휑(푈 ∩푓−1(푉 )) is necessarily open. Of course we can do it differently by not requiring 푓 to be continuous a priori but instead ask the above set to be open. For any charts ̃휑, 휓,̃ 휓̃ ∘ 푓 ∘ ̃휑−1 = (휓̃ ∘ 휓−1) ∘ (휓 ∘ 푓 ∘ 휑−1) ∘ (휑 ∘ ̃휑−1) is a composition of smooth map so is smooth. Thus smoothness of a map is independent of charts.

4 1 Manifolds

Definition (diffeomorphism). A smooth map 푓 ∶ 푀 → 푁 is a diffeomor- phism if 푓 is bijective and 푓−1 is smooth. If such 푓 exists, 푀 and 푁 are diffeomorphism.

Remark. 1. Our definition of smoothness is a generalisation of that in calculus. More precisely, 푓 ∶ R푛 → R푚 is smooth if and only if it is smooth in the calculus sense.

2. Any chart 휑 ∶ 푈 → R푑 is a diffeomorphism onto its image. 3. Composition of smooth maps is smooth.

5 2 Matrix Lie groups

2 Matrix Lie groups

2 We can view GL(푛, R) as an array of numbers and thus embed it in R푛 . Fur- thermore, by considering the determinant function it is an open subset so is an 푛2-manifold. Matrix multiplication is obviously smooth. In the same vein we have GL(푛, C), SL(푛, R) etc as mannifolds with smooth multiplications.

Definition (Lie group). A group 퐺 is a Lie group if 퐺 is a manifold and has a compatible group structure, i.e. the map (휎, 휏) ↦ 휎휏 −1 is smooth.

Before Lie groups, let’s have a short digression in analysis to discuss the exponential map. For a complex 푛 × 푛 matrix 퐴 = (푎푖푗), define a norm

|퐴| = 푛 ⋅ max |푎푖푗| 푖푗 where the 푛 in front is such that |퐴퐵| ≤ |퐴| ⋅ |퐵|. We now define 1 1 exp(퐴) ∶= 퐼 + 퐴 + 퐴2 + ⋯ + 퐴푛 + … 2 푛! Of course we have to check the series makes sense: in fact exp 퐴 is absolutely 퐴푛 |퐴|푛 convergent for all 퐴 as ∣ 푛! ∣ ≤ 푛! . The series is also uniformly convergent on any compact set, by Weierstrass 푀-test. Therefore exp is a well-defined continuous map. In fact, the map is smooth although the proof is technical. A sketch of the proof is like this: note that 푓 ∶ 퐴 → 퐴푚 is smooth for all 푚 ∈ N. For 푚 = 2, 푑푓퐴 ∶ 퐻 ↦ 퐻퐴 + 퐴퐻 so ‖푑푓퐴‖ ≤ 2|퐴| so for all 푚,

푚−1 ‖푑푓퐴‖ ≤ 푚|퐴|

We can thus term-by-term differentiate exp and get a locally uniformly conver- gent series and use the above estimate to bound derivatives. It is easy to check that: 1. exp(퐴푡) = (exp 퐴)푡 where 퐴푡 is the transpose of 퐴. 2. exp(퐶퐴퐶−1) = 퐶(exp 퐴)퐶−1 for 퐶 nonsingular.

3. exp(퐴 + 퐵) ≠ exp 퐴 exp 퐵 unless 퐴퐵 = 퐵퐴. 4. exp 퐴 exp(−퐴) = 퐼 for any matrix 퐴. The second property prompts us to put 퐴 into Jordan normal form before computing exp 퐴. Using the series

퐴2 퐴푛 log(퐼 + 퐴) = 퐴 − + ⋯ + (−1)푛+1 + … 2 푛 we can tackle this similarly to exp to check log(퐼 + 퐴) is smooth on {|퐴| < 1}. We can also check exp(log 퐴) = 퐴

6 2 Matrix Lie groups if |퐴 − 퐼| < 1, with a proof given by manipulation of double indexed series, which is valid due to absolute convergence. The expression log(exp 퐴) is more subtle. Clearly we need | exp 퐴 − 퐼| < 1. But it is not sufficient: if

0 −휃 퐴 = ( ) 휃 휃 0 where 휃 ∈ R, then cos 휃 − sin 휃 exp(퐴 ) = ( ) 휃 sin 휃 cos 휃 Put 휃 = 2휋, exp 퐴 − 퐼 = 0 but

log(exp 퐴2휋) = 0 ≠ 퐴2휋.

The reason is that the series is no longer absolutely convergent. If we add the additional condition that |퐴| < log 2 then

log(exp 퐴) = 퐴.

It is left as an exercise. (Hint: | exp |퐴| − 1| < 1 implies absolute convergence.) Example (orthogonal group). Recall that

푂(푛) = {퐴 ∈ GL(푛, R), 퐴퐴푡 = 퐼}. Let 퐴 ∈ 푂(푛) with |퐴−퐼| < 1. Let 퐵 = log 퐴 so 푒퐵 = 퐴. There exists 0 < 휀 < 1 such that whever |퐴 − 퐼| < 휀, have |퐵| < log 2 using continuity of log. Then

푒퐵푒퐵푡 = 퐴퐴푡 = 퐼 so 푒퐵 = 퐴 = (퐴푡)−1 = (푒퐵푡 )−1 = 푒−퐵푡 . Now |퐵푡| = |퐵| = log 2. Taking log, we find that 퐵 = −퐵푡 so 퐵 is a skew- symmetric matrix. Conversely, if 퐵 = −퐵푡, |퐵| < log 2 then

(푒퐵)푡 = 푒퐵푡 = 푒−퐵 = (푒퐵)−1 so 퐴 = 푒퐵 ∈ 푂(푛).

Proposition 2.1. 푂(푛) has a 퐶∞ structure making it a manifold and Lie 푛(푛−1) group of dimension 2 .

Proof. Put 푉0 ∶= {퐵 ∶ 퐵 skew-symmetric, |퐵| < log 2} and 푈 ∶= exp(푉0), an open neighbourhood of 퐼 ∈ 푂(푛). Let

ℎ ∶ 푈 → 푉0 퐴 ↦ log 퐴 which is a well-defined homeomorphism onto 푉0, an open subset of the skew- symmetric matrices, which can be identified with R푛(푛−1)/2.

7 2 Matrix Lie groups

Now we construct the charts (푈퐶, ℎ퐶). For all 퐶 = 푂(푛), put 푈퐶 ∶= {퐶퐴 ∶ 퐴 ∈ 푈}, i.e. left translation of 푈 by 퐶. Define

ℎ퐶 ∶ 푈퐶 → 푉0 퐴 ↦ log(퐶−1퐴) which is a homeomorphism . To check they form an atlas, first note that 퐶 ∈ 푈퐶 푂(푛) = ⋃ 푈 so 퐶∈푂(푛) 퐶. Furthermore

ℎ ∘ ℎ−1(퐵) = ℎ (퐶 푒퐵) = log(퐶−1퐶 푒퐵) 퐶2 퐶1 퐶2 1 2 1 which is smooth since it is the composition of smooth maps. Thus 푂(푛) is a manifold. To check compatibility of group axioms, define

퐹 ∶ 푂(푛) × 푂(푛) → 푂(푛) −1 (퐴1, 퐴2) ↦ 퐴1퐴2

In local coordinates, it is

−1 −1 ℎ −1 (퐹 (ℎ (퐵 ), ℎ (퐵 ))) 퐴1퐴2 퐴1 1 퐴2 2

−1 −1 퐵1 퐵2 −1 = log[(퐴1퐴2 ) 퐴1푒 (퐴2푒 ) ]

퐵1 −퐵2 −1 = log(퐴2푒 푒 퐴2 ) which is smooth.

The same construction works for other classical groups of matrices. See example sheet 1 Q4.

8 3 Tangent space to manifolds

3 Tangent space to manifolds

Consider a curve in R푛, defined by a smooth parameterisation 푛 푥(푡) = (푥푖(푡))푖=1 such that 푥(0) = 푝 ∈ R푛. Then a tangent to the curve is velocity 푛 푛 ̇푥(푡)∈ 푇푝R ≅ R . Let 푦 = 푦(푥) be a 퐶∞ change of variables, i.e. a local coordinates. Then

푛 푛 푑 퐷푦 휕푦푖 ∣ 푦(푥(푡)) = (푝) ̇푥(0)= (∑ 푎푗) 푑푡 푡=0 ⏟퐷푥 푗=1 휕푥푗 푖=1 Jacobian

Definition. A tangent vector 푎 to a manifold 푀 at a point 푝 ∈ 푀 is the 푛 assignment to each chart (푈, 휑), 푝 ∈ 푈 a 푛-tuple (푎1, … , 푎푛) ∈ R where 푛 = dim 푀 so that for another chart (푈 ′, 휑′), 푝 ∈ 푈 ′ with local coordinates ′ ′ (푥1, … , 푥푛) and (푥1, … , 푥푛), we have

푛 ′ ′ 휕푥푖 푎푖 = ∑ (푝)푎푗. 푗=1 휕푥푗

This is sometimes known as the tensorial definition of tangent space. Other equivalent definitions involve flows or using derivation on germs of smooth func- tions.

Definition (tangent space). The tangent space 푇푝푀 is the set of all tangent vectors to 푀 at 푝.

푛 It follows that 푇푝푀 is an 푛-dimensional real vector space. Thus 푇푝푀 ≅ R although this isomorphism is not canonical. But given a local coordinate chart with coordinates (푥1, … , 푥푛), the tuple (0, … , 1, … , 0) with 1 in 푖th position in 푛 has image under isomorphism 휕 (푝). Recall the chain rule for partial R 휕푥푖 derivatives: 푛 휕 휕푥푖 휕 ′ = ∑ ′ 휕푥푗 푖=1 휕푥푗 휕푥푖 ′ ′ where (푥1, … , 푥푛) is another chart. This is precisely the reason we impose upon tangent vectors the transformation rule, namely so that tangent vectors become a well-defined derivation of smooth functions at 푝: let 푎 = ∑ 푎 휕 (푝) ∈ 푇 푀 푖 푖 휕푥푖 푝 where 푥푖’s are local coordinates around 푝. Then a first order derivation at 푝 is

푎 ∶ 퐶∞(푀) → R 휕푓 푓 ↦ ∑ 푎푖 푖 휕푥푖 (where we assume the same charts on RHS) is a well-defined map independent of choice of coordinates. We can interpret, with the 푥푖’s, 푑 푎(푓) = ∣ 푓(푥(푡)) 푑푡 푡=0

9 3 Tangent space to manifolds for all 푥 ∶ (−휀, 휀) → 푀 smooth, 푥(0) = 푝 and ̇푥(0)= 푎. Now for another choice 푖̃푥 of local coordinates,

푑 휕푓 ̇ 휕푓 휕푗 ̃푥 ∣ 푓( ̃푥(푡))= ∑ (푝) 푗̃푥 (0) = ∑ (푝) (푝)푖 ̇푥 (0) 푑푡 푡=0 푗 휕푗 ̃푥 푗,푖 휕푗 ̃푥 휕푥푖 by the transformation law for tangent vectors. The derivations satisfy Leibniz rule, i.e.

푎(푓푔) = 푎(푓)푔(푝) + 푓(푝)푎(푔).

Conversely, every linear map 푎 ∶ 퐶∞(푀) → R satisfying the Leibniz rule arises from some 푎 ∈ 푇푝푀. This is left as an exercise. Example. An example from classical differential geometry. Consider a surface 푟 = 푟(푢, 푣) ∶ 퐷 → 푆 where 퐷 ⊆ R2 and 푆 = 푟(퐷) ⊆ R3. Then 푆 is a manifold −1 휕 휕 with 휑 = 푟 as a chart. Then 푟푢, 푟푣 at 푝 ∈ 푆 corresponds to 휕푢 , 휕푣 in our theory.

3.1 Lie algebra For a Lie group, the tangent spaces get an “infinitesimal” version of the group multiplication.

Definition. A Lie algebra is a vector space with a bilinear multiplication [⋅, ⋅], i.e. a Lie bracket such that 1. anticommutativity: [푎, 푏] = −[푏, 푎], 2. Jacobi identity: [[푎, 푏], 푐] + [[푏, 푐], 푎] + [[푐, 푎], 푏] = 0.

Theorem 3.1. Let 퐺 be a Lie group of 푛 × 푛 (real or complex) matrices such that log defines a coordinate chart near 퐼 ∈ 퐺, i.e. the image of log 푛2 near 퐼 is an open set in some real vector subspace of R . Identify 픤 = 푇퐼퐺 with the above open subset. Then 픤 is a Lie algebra with

[퐵1, 퐵2] ∶= 퐵1퐵2 − 퐵2퐵1

for 퐵1, 퐵2 ∈ 픤.

Proof. Check that 픤 is a vector space and [⋅, ⋅] is anticommutative. The Jacobi identity holds for matrices (straightforward check). What is left is to show 퐵1, 퐵2 ∈ 픤 then [퐵1, 퐵2] ∈ 픤. Consider

퐴(푡) = exp(퐵1푡) exp(퐵2푡) exp(−퐵1푡) exp(−퐵2푡), the commutator of two elements in 퐺. Then 퐴(0) = 퐼. Expand exp, we get

2 2 퐴(푡) = 퐼 + [퐵1, 퐵2]푡 + 표(푡 ) as 푡 → 0 so 2 2 퐵(푡) = log 퐴(푡) = [퐵1, 퐵2]푡 + 표(푡 ).

10 3 Tangent space to manifolds

In addition exp 퐵(푡) = 퐴(푡) holds for |푡| sufficiently small so 퐵(푡) ∈ 픤 as it is 퐵(푡) in the image of the log chart. It follows that 푡2 ∈ 픤 for 푡 ≠ 0 as 픤 is a vector space. Thus 퐵(푡) [퐵1, 퐵2] = lim ∈ 픤 푡→0 푡2 as every vector subspace of matrix 푛, C is a closed subset. Example. For 퐺 = 푂(푛), have 픤 = 픬(푛) = {skew-symmetric 푛 × 푛 matrices} by previous work.

Definition. 픤 is called the Lie algebra of 퐺, write 픤 = Lie(퐺).

In fact we can show Lie is a functor but we won’t pursue in that direction.

Definition (tangent bundle). Let 푀 be a smooth manifold. Then 푇 푀 = ∐ 푇 푀 푀 푝∈푀 푝 is the tangent bundle of .

Theorem 3.2. 푇 푀 has a natural 퐶∞ structure, making it into a smooth manifold of dim 푇 푀 = 2 dim 푀.

Proof. We shall induce the topology from the 퐶∞ structure. Let (푈, 휑) be a 푀 푈 = ∐ 푇 푀 푇 푀 = ⋃ 푈 푎 ∈ 푇 푀, 휑(푝) = chart on . Consider 푇 푝∈푈 푝 so 푇. For 푝 (푥 , … , 푥 ) so that 푎 = ∑ 푎 휕 . Now define 1 푛 푖 푖 휕푥푖

푛 푛 휑푇 ∶ 푈푇 → R × R

푎 ↦ (휑(푝), (푎푖))

To show compatibility, suppose (푈 ′, 휑′) is another chart on 푀 with local ′ ′ coordinates 푥푖 and efine ̃휑 as above. Then

′ −1 ′ ′ 휑푇 ∘ 휑푇 (푥, 푎) = (푥 , 푎 ) where 푥′ = 휑′ ∘ 휑−1(푥) and 푎′ is given by the translation law

′ ′ 휕푥푖 푎 = ∑ (푥)푎푗, 푗 휕푥푗 so is smooth wherever defined. Hausdorffness and second countability follows from that 푀 and R푛 are man- ifolds.

Note. Some remarks on the final statement regarding topological properties: 1. 푀 is 휎-compact, i.e. every open cover has a countable subcover.

2. A basis of topology of 푇 푀 is given by {퐵1 × 퐵2} where 퐵1 is open in 푛 some coordinate neighbourhood 푈 ⊆ 푀 and 퐵2 is open in R .

11 3 Tangent space to manifolds

Corollary 3.3. The projection

휋 ∶ 푇 푀 → 푀 (푝, 푎) ↦ 푝

is smooth. Remark. 푇 푀 has locally a product structure but in general 푇 푀 is not diffeo- morphic to 푀 × R푛.

Definition (vector field). A vector field on a manifold 푀 is a smooth map 푋 ∶ 푀 → 푇 푀 such that 휋 ∘ 푋 = id푀, i.e. 푋(푝) ∈ 푇푝푀 for all 푝 ∈ 푀.

Note that 푋 is smooth at 푝 if and only if for any coordinate neighbourhood 푈 푝 (푥 , … , 푥 ) 푋 = ∑푛 푎 (푥) 휕 푎 ∈ 퐶∞(푈) of with local coordinates 1 푛 , 푖=1 푖 휕푥 where 푖 for all 푖. Example. Every manifold has at least one vector field: sending every point to 0. This is not the most interesting example, however.

Theorem 3.4. Suppose dim 푀 = 푛. Then there exists smooth vector fields 푋(1), … , 푋(푛) on 푀 such that for all 푝 ∈ 푀, 푋(1)(푝), … , 푋(푛)(푝) is a basis 푛 of 푇푝푀, then 푇 푀 is isomorphic to 푀 × R .

Here “isomorphic” means that there is a diffeomorphism Φ ∶ 푇 푀 → 푀 × R푛 Φ| ∶ 푇 푀 → {푝} × 푛 such that 푇푝푀 푝 R is a linear isomorphism.

Definition (parallelisable). A manifold satisfying the hypothesis is called parallelisable.

푝 ∈ 푀 휋(푎) = 푝 푎 ∈ 푇 푀 푎 = ∑푛 푎 푋(푖)(푝) Proof. Consider so for all 푝 . Then 푖=1 푖 for some unique 푎푖 ∈ R. Put

푛 Φ(푎) ∶= (휋(푎), (푎1, … , 푎푛)) ∈ 푀 × R Φ| which is clearly a bijection and 푇푝푀 is a linear isomorphism. Thus suffices to −1 check Φ is a diffeomorphism. We use chart (푈, 휑) on 푀 and let (휋 (푈), 휑푇) be the corresponding chart on 푇 푀. Then

−1 푛 푛 (휑, 푛 ) ∘ Φ ∘ 휑 ∶ (푥, (푏 ) ) ↦ (푥, (푎 ) ) idR 푇 푖 푖=1 푖 푖=1 such that (푖) 휕 푎 = ∑ 푎푖푋 (푝) = ∑ 푏푗 (푝). 푖 푗 휕푥푗

It is then obvious that these 푎푖’s and 푏푗’s differ by a change-of-basis transfor- mation. Explicitly, write 푋(푖) in local basis as

(푖) (푖) 휕 푋 |푈 = ∑ 푋푗 (푥) 푗 휕푥푗

12 3 Tangent space to manifolds and so (푖) 푏푗 = ∑ 푎푖푋푗 (푥) 푖 (푖) so Φ is smooth. To check the inverse, note that (푋푗 (푥)) is a non-singular matrix smooth in 푥 so Φ−1 also has a smooth local expression.

Remark. 1. The converse is easily seen to be true, in which case vector fields on 푀 are simply 퐶∞(푀, R푛). 2. The parallelisable hypothesis is quite restrictive. For example, it implies that each 푋(푖) is never-zero. Some manifolds, such as 푆2 do not have such vector fields at all. In fact, 푆푛 is parallelisable if only if 푛 = 1, 3, 7. 3. Every orientable 3-dimensional vector field is parallelisable.

Exercise. Show 푆2푛+1 has a never-zero vector field.

Definition (differential). Let 퐹 ∶ 푀 → 푁 be a smooth map. Then the differential of 퐹 at 푝 ∈ 푀 is a linear map

푑퐹푝 ∶ 푇푝푀 → 푇퐹(푝)푁

such that if 푥푖 is a local coordinate near 푝 and 푦푖 is a local coordinate near 퐹 (푝), then ̂ 휕 휕퐹푗 휕 푑퐹푝 ∶ ↦ ∑ (푥(푝)) (퐹 (푝)) 휕푥푖 푗 휕푥푖 휕푦푗 where 퐹̂ = 휓 ∘ 퐹 ∘ 휙−1 is the coordinate representation of 퐹.

Now we have to check that it is independent of coordinate representation. Recall that 휕 휕푥푖 ′ 휕 ′ (푝) = ∑ ′ (푥 (푝)) (푝) 휕푥푘 푖 휕푥푘 휕푥푖 and similar for 휕 (퐹 (푝)). So 휕푦푗

휕 휕푥 휕푦 휕푦′ 휕 푑푓 ∶ (푝) ↦ ∑ 푖 푗 ℓ (퐹 (푝)) 푝 휕푥′ 휕푥′ 휕푥 휕푦 휕푦′ 푘 푖,푗,ℓ ⏟⏟⏟⏟⏟푘 푖 푗 ℓ 휕푦′ ℓ 휕푥′ by chain rule 푘 so 푑퐹푝 is indeed invariantly defined.

퐹 퐺 Exercise (geometer’s chain rule). For 푀 −→푁 −→푍, we have

푑(퐺 ∘ 퐹 )푝 = 푑퐺퐹(푝) ∘ 푑퐹푝.

Now suppose 퐹 ∶ 푀 → 푁 is a diffeomorphism and 푋 is a vector field on 푀. Then (푑퐹 )푋 is a valid vector field on 푁.

13 3 Tangent space to manifolds

Remark. In general a vector field does not admit a pushforward. Weneed surjectivity so (푑퐹 )푋 is everywhere defined and injectivity to avoid conflicting values on target points. Finally we need the inverse to be smooth to ensure (푑퐹 )푋 is smooth. Every vector field 푋 defines a linear map from 퐶∞(푀) to itself. More precisely, it is a first order derivation with 푝 ∈ 푀 varying. Locally, if 푋 = ∑ 푋 (푥) 휕 then 푋ℎ is given by 푖 푖 휕푥푖 휕ℎ 푋ℎ = ∑ 푋푖(푥) (푥) 푖 휕푥푖 It is an easy exercise to check that it is invariantly defined. On the other hand, a smooth function on 푁 can always be pulled back by 퐹. Suppose 푓 ∈ 퐶∞(푁). Then 푓 ∘ 퐹 ∈ 퐶∞(푀). Thus in any local coordinates 푥푖 on 푀, 푦푖 on 푁, have 휕 휕푓 휕푦 (푝)(푓 ∘ 퐹 ) = ∑ (푦(퐹 (푝)) 푗 (푥(푝)). 휕푥푖 푗 휕푦푗 휕푥푖

Thus we have a coordinate-free formula

푋(푓 ∘ 퐹 ) = ((푑퐹 )푋푓) ∘ 퐹 .

Equivalently, the following diagram commutes:

∗ 퐶∞(푁) 퐹 퐶∞(푀)

(푑퐹)푋 푋 ∗ 퐶∞(푁) 퐹 퐶∞(푀)

Let 푋 and 푌 be two vector fields on 푀 considered as first order linear differ- ential operators. The composition 푋푌 is not a vector field. However,

푍 ∶= [푋, 푌 ] ∶= 푋푌 − 푌 푋 is a vector field. In local coordinates, it is 휕푌 휕푋 휕 ∑ (푋 푘 − 푌 푘 ) 푖 휕푥 푖 휕푥 휕푥 푖,푘 푖 푖 푘 where 푋 = ∑ 푋 휕 , 푌 = ∑ 푌 휕 . Check that second order derivatives 푖 푖 휕푥푖 푖 푖 휕푥푖 vanish (because of symmetry of mixed partials). We can check that 푍 is a vector field and as a map 퐶∞(푀) → 퐶∞(푀) it is linear over R and satisfy 푍(푓푔) = (푍푓)푔 + 푓푍푔.

Thus vector fields on 푀 form a Lie algebra, which is infinite-dimensional. Denote it by 푉 (푀).

14 3 Tangent space to manifolds

3.2 Left-invariant vector field

Let 퐺 be a Lie group and 푒 ∈ 퐺 the identity element. Let 픤 = 푇푒퐺. Given 푔 ∈ 퐺, the left translation by 푔

퐿푔 ∶ 퐺 → 퐺 ℎ ↦ 푔ℎ is a smooth map and since it has inverse 퐿푔−1 , it is a diffeomorphism. Given 휉 ∈ 픤, we can define a vector field by

푋휉(푔) ∶= (푑퐿푔)푒(휉) ∈ 푇푔퐺.

We can do this because for every point there is a diffeomorphism sending 푒 to it, and therefore we can construct a vector field using only its information at 푒. We will soon find out there are lots of symmetry involved.

Lemma 3.5. The map 푋휉 ∶ 퐺 → 푇 퐺, 푔 ↦ (푑퐿푔)푒(휉) is smooth so 푋휉 is a smooth vector field.

Proof. As usual, check smoothness in local coordinates. Consider group multi- plication 퐿 ∶ 퐺 × 퐺 → 퐺. Fix 푔0 ∈ 퐺, then around (푔0, 푒) ∈ 퐺 × 퐺, given charts 휑 푒 푉 휑 푔 푉 푒 around whose image is 푒 and 푔0 around 0 whose image is 푔0 , the local expression 퐿̂ is

푉 × 푉 → 푉 ′ 푔0 푒 푔0 퐿̂ = 휑 (퐿(휑−1(⋅), 휑−1(⋅))) 푔0 푔0 푒

퐿̂ = 퐿(휑̂ (푔), ⋅) ∶ 푉 → 푉 ′ 퐷 퐿̂ 푉 Then 푔 푔0 푒 푔0 .Thus 2 , the derivative of the 푒 variables, ̂ gives the coordinate expression for (푑퐿푔)푒. But 퐷2퐿 depends smoothly on the 푉 퐿 퐶∞ 푋 푔 푔0 variables as is a map. Therefore 휉 is a smooth map around 0. As 푔0 is arbitrary 푋휉 is smooth.

We have shown that, identifying 픤 = 푇푒퐺, that

(푑퐿푔)푒 ∶ 픤 → 푇푔퐺 is smooth. In fact, it is a linear isomorphism. Thus we have

Proposition 3.6. If 휉1, … , 휉푛 are linearly independent (form a basis, respec- 픤 푔 ∈ 퐺 푋 (푔), … , 푋 (푔) tively) in then for all , 휉1 휉푛 are linearly independent (form a basis, respectively) in 푇푔퐺.

As a consequence we have

Theorem 3.7. Every Lie group 퐺 is parallelisable, i.e. 푇 퐺 ≅ 퐺 × Rdim 퐺 where the diffeomorphism restricts to 푇푔퐺 is an isomorphism onto {푔} × Rdim 퐺 for all 푔 ∈ 퐺.

15 3 Tangent space to manifolds

There is another symmetry for a Lie group 퐺. For all 푔, ℎ ∈ 퐺, for all 휉 ∈ 픤,

(푑퐿푔)ℎ푋휉(ℎ) = (푑퐿푔)ℎ(푑퐿ℎ)푒휉 = (푑퐿푔ℎ)푒휉 = 푋휉(푔ℎ), i.e. (푑퐿푔)푋휉 = 푋휉 ∘ 퐿푔, (∗) which, if we view 푋휉 ∶ 푀 → 푇 푀 as a global section of the projection map, may also be written as (푑퐿푔)푋휉 = 푋휉.

Definition (left-invariant vector field). A vector field 푋 on a Lie group satisfying the (∗) is called left-invariant. Denote the subspace of all left- invariant vector fields by ℓ(퐺).

One important observation is that ℓ(퐺) is a finite-dimensional subspace of 푉 (퐺): it is easy to see that for all 푋 ∈ ℓ(퐺), there exists a unique 휉 ∈ 픤 such that 푋 = 푋휉. This induces an isomorphism ℓ(퐺) ≅ 픤 so dim ℓ(퐺) = dim 퐺. Even better, ℓ(퐺) is closed under Lie bracket so

Theorem 3.8. ℓ(퐺) is a Lie subalgebra of 푉 (퐺).

Proof. One way to show this is to use 푋(푓 ∘ 퐹 ) = ((푑퐹 )푋푓) ∘ 퐹

∞ with 퐹 = 퐿푔, 푋 = 푋휉 and 푓 ∈ 퐶 (퐺). See example sheet. Alternatively, for any vector fields 푋, 푌 and diffeomorphism 퐹, in example sheet 1 Q6 we show that (푑퐹 )[푋, 푌 ] = [(푑퐹 )푋, (푑퐹 )푌 ].

Let 푓 ∈ 퐶∞(퐺), 푔 ∈ 퐺, 휉, 휂 ∈ 픤. Then

((푑퐿푔)[푋휉, 푋휂]푓) ∘ 퐿푔

=([(푑퐿푔)푋휉, (푑퐿푔)푋휂]푓) ∘ 퐿푔

=([푋휉 ∘ 퐿푔, 푋휂 ∘ 퐿푔]푓) ∘ 퐿푔

=(([푋휉, 푋휂] ∘ 퐿푔)푓) ∘ 퐿푔 which is saying (푑퐿푔)[푋휉, 푋휂] = [푋휉, 푋휂] ∘ 퐿푔 which is precisely (∗). We now have, in the case of “good” matrix Lie groups, two definitions making 픤 into a Lie algebra. In fact, they are equivalent.

Theorem 3.9. Let 퐺 be a matrix Lie group with log defining a chart around 푒 ∈ 퐺. Then

푇푒퐺 → ℓ(퐺)

휉 ↦ 푋휉

is an isomorphism of the Lie algebras (using the “matrix” definition on

16 3 Tangent space to manifolds

LHS). We will prove the theorem in the next chapter.

17 4 Submanifolds

4 Submanifolds

Let 푀 be a manifold, 푁 ⊆ 푀 and 푁 is itself a manifold (not a priori to be have restriction of smooth structure on 푀). Denote 휄 ∶ 푁 → 푀 the inclusion map.

Definition (embedded submanifold). If 휄 is smooth, 푑휄푝 ∶ 푇푝푁 → 푇푝푀 is injective for all 푝 ∈ 푁 and 휄 is a homeomorphism onto its image, then we say 푁 is an embedded submanifold of 푀.

Definition (immersed submanifold). If we drop the requirement that 휄 is a homeomorphism then 푁 is a immersed submanifold of 푀. At this point, the topological requirement may seem a bit mysterious and not clear what it entails. It is equivalent to the statement that 퐷 ⊆ 푁 is open in 푁 if and only if 퐷 = 푈 ∩ 푁 for some 푈 ⊆ 푀 open. It excludes situation like mapping an open interval to figure 8. A variant of the definition may omit 푁 ⊆ 푀 and instead require 휓 ∶ 푁 → 푀 to be an embedding. If 휓 is injective and the three properties hold then 휓(푁) ⊆ 푀 is an embedded submanifold. Convention. From now on submanifold means by default embedded subman- ifold. Notation. For manifolds 푀 and 푁 and 휓 ∶ 푁 → 푀 an embedding, write 휓 ∶ 푁 ↪ 푀, i.e. 휓(푁) is a submanifold of 푀. Remark. An immersion 휓 ∶ 푁 → 푀 means that 푑휓 is injective everywhere. In this way we may obtain 휓(푁) ⊆ 푀 “immersed with self-intersections”. Example. 1. For curves and surfaces in R3, 훾 ∶ (0, 1) → R3 and 푟 ∶ 푈 → R3 where 푈 ⊆ R2, the conditions mean that (a) 훾 and 푟 are smooth, (b) they are regular parameterisations, (c) depending on what we are interested in, we may require them to be (topological) embeddings. 2. The figure 8 example mentioned to distinguish embedded vs. immersed manifold may seem contrived. However, immersed (and not embedded) manifolds emerge naturally from irrational twist flow. Consider the map

R → 푆1 × 푆1 푡 ↦ (푒푖푡, 푒푖훼푡)

where 훼 ∈ R \ Q. We can check that the map is injective and the image is dense in 푆1 × 푆1. In particular it is not a toplogical emedding so this is an immersion but not an embedding. One frequently asked question is: is a submanifold of R푛 the same as 푓−1(0) for some smooth map 푓 ∶ R푛 → R푘? In general, no! 푓−1(0) ⊆ R푛 is closed. One can check that for every closed 퐸 ⊆ R2 there exists a smooth 푓 ∶ R2 → R such that 푓−1(0) = 퐸.

18 4 Submanifolds

Definition (regular value). Let 푓 ∶ 푀 → 푌 be a smooth map. 푞 ∈ 푌 is a regular value of 푓 if for all 푝 ∈ 푀 such that 푓(푝) = 푞, 푑푓푝 is surjective.

Note that under this definition if 푞 ∉ 푓(푀) then 푞 is vacuuously a regular value of 푓.

Theorem 4.1. Let 푓 ∶ 푀 → 푌 be a smooth map and 푞 ∈ 푌 is regular value of 푓. If 푓−1(푞) ≠ ∅ then 푁 = 푓−1(푞) is an embedded submanifold with

dim 푁 = dim 푀 − dim 푌 . This is geometer’s implicit function theorem, also known as preimage theorem. We assume this theorem without proof, which can be found in the lecturer’s online notes. Remark. By a result in differential topology, suppose 푀 is a manifold and 푁 ⊆ 푀 is equipped with subspace topology. If there exists a smooth structure on 푁 such that 푁 ⊆ 푀 is submanifold then this structure is unique. Thus it makes sense to say 푁 is or isn’t a submanifold of 푀.

Proposition 4.2. Let 푁 ↪ 푀 and 푝 ∈ 푁. Then there eixsts a neighbour- hood 푈 of 푝 in 푀 and a smooth map 푓 ∶ 푈 → R푑 where 푑 = dim 푀 − dim 푁 such that 푁 ∩ 푈 = 푓−1(0). 푑 is called the codimension of 푁 in 푀.

푛 Proof. Let 휑 ∶ 푈0 → R is a chart on 푀 where 휑(푝) = 0 with local coordinates ℓ (푥1, … , 푥푛). Let 휓 ∶ 푉0 → R be a chart on 푁 where 휓(푝) = 0 with local coordinates (푦1, … , 푦ℓ). Then the inclusion 휄 ∶ 푁 → 푀 has local exression

푥푖 = 푥푖(푦).

The derivative at 0 휕푥 ( 푖 (0)) 휕푦 푗 푛×ℓ has rank ℓ. wlog we may assume the top ℓ × ℓ submatrix is nonsingular. Then by Inverse function theorem,

푦푗 = 푦푗(푥1, … 푥ℓ) which is well-defined and smooth near 0. Then for 푖 > ℓ,

푥푖 = 푥푖(푦(푥1, … , 푥ℓ)) = ℎ푖(푥1, … , 푥ℓ).

Then 푓푖(푥) ∶= 푥푖 − ℎ푖(푥1, … , 푥ℓ) for 푖 > ℓ gives a required 푓 ∶ 푈 → R푑 where 푑 = 푛 − ℓ, with Jacobian 1 0 휕푓 = ⎜⎛ ∗ ⋮ ⎟⎞ 휕푥 ⎝ 0 1⎠ so a regular value.

19 4 Submanifolds

2 This result cannot be improved: let 푀 = R푃 and 푁 = {푥0 ∶ 푥1 ∶ 푥2 ∈ 2 1 1 −1 R푃 ∶ 푥2 = 0} which can be identified with R푃 ≅ 푆 . But 푁 ≠ 푓 (푞) for all smooth 푓 ∶ R푃 2 → 푃 where 푃 is a one-dimensional manifold, with 푞 a regular value. A sketch of proof: if there exists such an 푓 then there exsits some chart 휓 on 푈 around 푞, 휓 ∘ 푓 ∶ 푈 → (−1, 1) and 푈 ⊇ 푁. Suppose for contradiction 푁 = {푝 ∶ (휓 ∘ 푓)(푝) = 0}. R푃 2 \ 푁 is homeomorphic to an open disk in R2. 휓 ∘ 푓 has 0 as a regular value, implying that 휓 ∘ 푓 takes both positive and negative values. But R푃 2 \ 푁 is connected so contradiction.

Theorem 4.3 (Whitney embedding theorem). Every 푛-dimensional man- ifold 푀 admits a smooth embedding to R2푛.

It is a hard theorem but it is very easy to show that there exists 푁 such that 푀 is a submanifold of R푁. In example sheet 1 Q9 we showed this for 푀 compact. It is also not too difficult to set 푁 = 2푛+1, basically by embedding it in a sufficiently large space and whittle down the dimension. However thelast step of improvement is truely an ingenious piece of work. The remarkability of this theorem is not to say that the intrinsic defintion of manifolds coincide with the extrinsic one, but the optimal dimension R2푛 of ambient space. This is a topological invariant measuring how “complicated” a geometric object is. For example, 푆푛 can be embedded in R푛+1 but R푝2 cannot be embedded in R3. The Klein bottle does not embed in R3 either. Restatement of an earlier theorem:

Theorem 4.4. Suppose 퐺 ⊆ GL(푛, C) is a subgroup and a Lie group with 퐶∞ structure of 퐺 given by log charts (i.e. log maps an open neighbourhood 푈퐼 of the idenity 퐼 onto a neighbourhood of 0 is some real subspace 푉0 of Mat(푛, C)), then 픤 → ℓ(퐺)

휉 ↦ 푋휉

is a Lie algebra isomorphism. Here 픤 = 푇퐼퐺 ⊆ Mat(푛, C) is the span of 푉0 over R. Proof. We have shown earlier

[푋휉, 푋휂] = 푋휁 for some 휁 ∈ 픤. Now want to show that

휁 = [휉, 휂] = 휉휂 − 휂휉 as matrices in 픤. First 퐺 = GL(푛) = Mat(푛) over R or C so 픤 = 푚푎푡(푛). Then for 푔 ∈ GL(푛), 퐿푔 is a linear map so (푑퐿푔)ℎ for all 푔, ℎ in the usual matrix multiplication The local expression for 퐿푔 around 퐼 ∈ GL(푛) is 1 (퐿̂ )퐵 = 푔 ⋅ exp 퐵 = 푔 ⋅ (퐼 + 퐵 + 퐵2 + … ) 푔 2! so ̂ (푑퐿푔)0퐶 = 푔퐶

20 4 Submanifolds

푖 푖 Therfore for all 푔 = (푔푗) ∈ GL(푛), 퐴 = (퐴푗) ∈ 픤 = 푇퐼퐺퐿(푛), we have

푖 휕 푖 푘 휕 푋퐴(푔) = ∑ 푋푗(푔) 푖 = ∑ 푔푘퐴푗 푖 . 푖,푗 휕푔푗 푖,푗,푘 휕푔푗

Now the chain rule follows from straightforard calculation. Using formula for Lie brackets of vector fields,

푖 푘 휕 ℓ 푝 푘 휕 ℓ 푝 휕 푖 푘 휕 푔푘 (퐴푗 푖 (푔푝퐵푞 ) − 퐵푗 푖 (푔푝퐴푞 )) ℓ = 푔푘(퐴퐵 − 퐵퐴)푗 ∘ 푖 . 휕푔푗 휕푔푗 휕푔푞 휕푔푘

For the general case 퐺 ⊆ GL(푛), note that the log chart hypothesis implies that 휄 ∶ 퐺 ↪ GL(푛). In fact we’ll use 푈퐼 ↪ GL(푛) where 푈퐼 ⊆ 퐺. For all 푔 ∈ 퐺, 퐿푔 ∶ 퐺 → 퐺 is the restriction of 퐿푔 ∶ GL(푛) → GL(푛). Furthermore for all ℎ ∈ 퐺, (푑퐿푔)ℎ ∶ 푇ℎ퐺 → 푇푔ℎ퐺 is the corresponding restriction of (푑퐿푔)ℎ on GL(푛). Therefore 푋휉 ∈ ℓ(퐺) is a restriction of 푋휉 ∈ ℓ(GL(푛)). Then

[푋휉|퐺, 푋휂|퐺] = [푋휉, 푋휂]|퐺 which can be verified in “adapted” local coordinates using local test functions only depending on coordinates along 퐺 and constant in the normal direction (graph) But on GL(푛) we know [푋휉, 푋휂] = 푋[휉,휂], also for all 휉, 휂 ∈ 픤, [휉, 휂] ∈ 픤. Now the theorem for 퐺 follows too.

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5 Differential forms

Definition (cotangent space). Suppose 푀 is an 푛-dimensional manifold and 푝 ∈ 푀. The dual of 푇푝푀, consisting of all linear funcitons 푇푝푀 → R, is call ∗ the cotangent space at 푝 and denoted by 푇푝 푀.

휕 If 푥푖’s are local coordinates then we have 휕푥 ∣ as a basis for 푇푝푀. The dual 푖 푝 ∗ basis of 푇푝 푀 is denoted by (d푥푖)푝, i.e.

휕 d푥푖( ) = 훿푖푗. 휕푥푗

∗ Therefore for all 푎 ∈ 푇푝 푀, we can express uniquely as

푛

푎 = ∑ 푎푖(d푥푖)푝. 푖=1

′ Recall transformation rule for tangent space: if 푥푖 is another coodinates then

휕 휕푥푘 휕 ′ = ∑ ′ 휕푥푖 휕푥푖 휕푥푘 so by linear algebra 휕푥푖 ′ d푥푖 = ∑ ′ d푥푗 휕푥푗 so if a 1-form is given in two local coordinates

′ ′ 푎 = ∑ 푎푖d푥푖 = ∑ 푎푗d푥푗 푖 푗 then we have the transformation law

′ 휕푥푖 푎푗 = ∑ ′ 푎푖 푖 휕푥푗 which is not the same as that for tangent space.

Definition (cotangent bundle). The cotangent bundle of a manifold 푀 is defined by ∗ ∗ 푇 푀 = ∐ 푇푝 푀. 푝∈푀

Theorem 5.1. 푇 ∗푀 is a smooth manifold of twice the dimension of that of 푀. Moreover the natural projection map 휋 ∶ 푇 ∗푀 → 푀 is smooth.

Proof. Similar to that of tangent bundle.

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Definition (differential 1-form). A (smooth) differential 1-form, also known as 1-form 훼 is a smooth map 훼 ∶ 푀 → 푇 ∗푀 that is a section of 휋 ∶ 푇 ∗푀 → ∗ 푀, i.e. 훼(푝) ∈ 푇푝 푀 for all 푝 ∈ 푀.

1 훼 = ∑ 훼 푑푥 훼 Similar to vector fields, a -form 푖 푖 푖 is smooth if and only if 푖 is smooth in all local coordinates, if and only if for all 푋 ∈ 푉 (푀), 훼(푋) ∈ 퐶∞(푀). To define general 푘-forms, we need to take a crash course in multilinear algebra. For 푟 = 0, 1, …, the 푟th exterior product

푟 ∗ 푟 Λ 푇푝 푀 = {alternating multilinear functions on (푇푝푀) }. It is a vector space with basis { 푥 ∧ … ∧ 푥 }, 1 ≤ 푖 < ⋯ < 푖 ≤ 푛 d 푖1 d 푖푟 1 푟 where 푛 = dim 푀 and 푥푖’s are local coordinates around 푝. They are defined by 푥 ∧ … ∧ 푥 (푣 , … , 푣 ) = (( 푥 (푣 )) ) d 푖1 d 푖푟 1 푟 det d 푖푘 ℓ 푘,ℓ

휕 휕 where 푣ℓ = 휕푥 and 푘ℓ ≤ 푘ℓ+1 for all ℓ. This equals to 1 if and only if 푣푘 = 휕푥 푘ℓ 푖푘 for all 푘 and 0 otherwise.

Notation (multiindex notation). Let 퐼 = (푖1, … 푖푟) be a multiindex, then write 푥 = 푥 ∧ … ∧ 푥 . d 퐼 d 푖1 d 푖푟 Using this notation, the transformation law for 푟-forms under change of coordi- ′ nates from 푥푖 to 푥푗 is 푟 휕푥′ d푥′ = ∑ ∏ 퐽푘 d푥 . 퐽 휕푥 퐼 퐼 푘=1 푖푘 Using the same method in the construction of smooth structure on 푇 푀 and 푇 ∗푀, together with the transformation law for 푟-forms, we can make

푟 ∗ 푟 ∗ Λ 푇 푀 = ∐ Λ 푇푝 푀 푝∈푀

푛 푟 ∗ into a manifold of dimension 푛 = (푟) with smooth projection 휋 ∶ Λ 푇 푀 → 푀. It is called the bundle of differential 푟-forms. Note. Λ0푇 ∗푀 = 푀 × R and Λ1푇 ∗푀 = 푇 ∗푀

Definition (differential form). A smooth differential 푟-form, or simply an 푟-form on 푀 is a smooth map 훼 ∶ 푀 → Λ푟푇 ∗푀 that is a section of 휋 ∶ Λ푟푇 ∗푀 → 푀. The space of all 푟-forms on 푀 is denoted Ω푟(푀).

Given local coordinates 푥푖’s, a differential form 훼 can be locally expressed as 훼 = ∑ 훼퐼(푥)d푥퐼 퐼 where 훼 ’s are smooth. They can be computed by 훼 = 훼( 휕 ). It can be easily 퐼 퐼 휕푥퐼 checked that 훼 is smooth if and only if 훼(푋1, … , 푋푟) is smooth for all smooth vector fields 푋1, … , 푋푟. Note that 0 forms are just smooth functions 푀 → R so Ω0(푀) = 퐶∞(푀).

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5.1 Orientation of manifolds

Theorem 5.2. For an 푛-manifold, TFAE: 1. there exists a never-zero 푛-form on 푀,

2. there exists a family of charts (휑훼, 푈훼) such that 푀 = ⋃ 푈훼 and for ′ ′ all 훼, 훼 with local coordinates 푥푖, 푥푖 on 푈훼, 푈훼′ respectively, we have

′ 휕푥푗 det > 0 휕푥푖

on 푈훼 ∩ 푈훼′ , 3. Λ푛푇 ∗푀 is parallelisable.

Proof. 1. 1 ⟹ 2: the transformation law when a differential form is expressed in ′ local coordinates 푥푖 and 푥푖 is

d푥1 ∧ … ∧ d푥푛 휕푥 휕푥 = (∑ 1 d푥′ ) ∧ … ∧ (∑ 푛 d푥′ ) 휕푥′ 푖1 휕푥′ 푖푛 푖1 푖1 푖푛 푖푛 = … =

푛 Suppose we have 휔 ∈ Ω (푀) never-zero. Let 푀 = ⋃ 푈훼 where each 푈훼 is open connected, then

휔| = 푓 (푥훼) 푥훼 ∧ … ∧ 푥훼 푈훼 훼 d 1 d 푛

훼 ′ We can reorder 푥푖 such that 푓훼 > 0. Then all deteminants are 푓훼/푓훼 > 0 2. 2 ⟹ 1: we need partition of unity. Given such a collection of charts, define on each 푈훼 a never-zero 푛-form

훼 훼 휔훼 ∶= d푥1 ∧ … ∧ d푥푛

Let {휌푖} be a partition of unity subordinate to the cover. Then

휔 = ∑ 휌 (휔 ) 푖 훼푖 푖

is a well-defined never-zero form on Ω푛(푀). 3. 1 ⟺ 3: this involves the same idea in the proof that the existence of global (co)frame is equivalent to parallisability of (co)tangent bundle. In fact, this generalises to any bundle. We omit the details.

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Theorem 5.3 (partition of unity). For every open cover 푀 = ⋃ 푈훼, there ∞ exists a countable collection 휌푖 ∈ 퐶 (푀) such that

1. for all 푖, supp 휌푖 is compact and is contaned in 푈훼 for some 훼.

2. locally finite: for every 푥 ∈ 푀, there exists an open neighbourhood 푊푥 푥 휌 | ≠ 0 푖 containing such that 푖 푊푥 only for finitely many ’s. 휌 ≥ 0 푖 ∑ 휌 = 1 3. 푖 for all and 푖 푖 .

Then {휌푖} is a partition of unity subordinate to {푈훼}. Proof. Omitted.

Definition (orientability). A manifold satisfying any of the condition is called orientable. If a manifold is orientable and connected then there are precisely two choices of orientation. The orientation is induced, equivalently, by 1. 휔 ∈ Ω푛(푀) up to a positive smooth function, 2. a coordinate cover up to “positive compatibility”, 3. a choice of 휙 up to composition (푥, 푎) ↦ (푥, ℎ(푥)푎) where ℎ ∈ 퐶∞(푀), ℎ > 0.

5.2 Exterior derivative Let 푀 be a smooth manifold and 푓 a smooth function on 푀. The differential at a point 푝 ∈ 푀 is a linear map 푑푓푝 ∶ 푇푝푀 → 푇푓(푝)R = R. Thus 푑푓푝 can be ∗ thought as an element of 푇푝 푀. Therefore for all 푋 ∈ 푉 (푀), have ∞ 푑푓푝(푋) = 푋푓(푝) ∈ 퐶 (푀). By checking coeficients of 푑푓 in local coordinates we find that 푑푓 ∈ Ω1(푀) is well-defined. 1 Notation. For all local coordiates 푥푖 defined on 푈 ⊆ 푀, d푥푖 = 푑푥푖 ∈ Ω (푈). Remark. This is what is called gradient in vector calculus on R푛. However, in vector calculus we identify R with its dual, which is not always the case for manifolds. Thus gradient is a covector, or 1-form.

Theorem 5.4 (exterior derivative). There is a unique R-linear map d ∶ Ω푘(푀) → Ω푘+1(푀) for all 푘 = 0, 1, … such that 1. if 푓 ∈ Ω0(푀) then d푓 is the differential 푑푓. 2. d(휔 ∧ 휂) = (d휔) ∧ 휂 + (−1)deg 휔휔 ∧ d휂 where deg 휔 = 푘 means that 휔 ∈ Ω푘(푀).

3. d2 = 0.

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d is called the exterior derivative.

Proof. Choose 푈 ⊆ 푀 with local coordinates 푥푖 and suppose 휔|푈 = 푓(푥)d푥퐼. Then

d(푓(푥)d푥퐼) 푟 푘+1 =(d푓) ∧ d푥퐼 + 푓(푥) ∑(… ∧ (−1) dd푥푘 ∧ … ) 푘=1 푛 휕푓 = ∑ (푥)d푥푖 ∧ d푥퐼 푖=1 휕푥푖 and extend to Ω푟(푈) by linearity. Then 1 clearly holds, 2 is an easy verification and 3 holds by symmetry of mixed partials. This shows the uniqueness of d. The above computation also shows that d must be a local operator, i.e. d휔푝 is determined by 휔|푈 for any coordinate neighbourhood 푈 of 푝. Thus for ′ existence of d it suffices to check if given another local coordinates 푥푖 on 푈, the two local expressions agree. Suppose d′ is the exterior derivative given by ′ ′ 푟 another coordinates 푥푖 on 푈. Aim to show d = d on Ω (푈) for all 푟. 푟 ′(푓 푥 ) = ′푓 ∧ 푥 + ∑(−1)푘+1푓 푥 ∧ … ∧ ′ 푥 ∧ … ∧ 푥 d d 퐼 d d 퐼 d 푖1 d d 푖푘 d 푖 푘=1 ′푓 = 푓 푓 0 ′푥 = 푥 푥 d d as is a -form and d 푖푘 d 푖푘 as 푖푘 are smooth functions. Finally, ′ 푥 = 푥 = 0 d d 푖푘 dd 푖푘 ′ ′ so d(푓d푥퐼) = d (푓d푥퐼). d = d .

5.3 Pullback of differential forms

Definition (pullback). Let 푓 ∶ 푀 → 푁 be a smooth map. Then the pullback induced by 푓 is

푓∗ ∶ Ω푟(푁) → Ω푟(푀) 훼 ↦ 푓∗(훼)

where ∗ (푓 (훼))푝(푣1, … , 푣푟) = 훼푓(푝)((푑푓푝)푣1, … , (푑푓푝)푣푟)

for all 푝 ∈ 푀, 푣푖 ∈ 푇푝푀.

Note. 1. Unlike vector fields which in general cannot be pushed forward unless 푓 is a diffeomorphism, 푓∗ is well-defined for all maps 푓. 푔 푓 2. Chain rule for pullbacks: given 푍 −→푀 −→푁 then (푓 ∘ 푔)∗ = 푔∗ ∘ 푓∗, i.e. pullback is a contravariant functor. 3. 푓∗(훼∧훽) = 푓∗(훼)∧푓∗(훽) which is a straightforward exercise. In particular for all ℎ ∈ 퐶∞(푁), 푓∗(ℎ훼) = (ℎ ∘ 푓)푓∗(훼) which implies that 푓∗ is 퐶∞(푀)-linear.

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4. Exterior derivative commutes with pullbacks, i.e. d(푓∗훼) = 푓∗(d훼). For a proof, suffices to check this on R푛 as d is local and 푓∗ is pointwise (i.e. algebraic). The outline is as follow: wlog 푀 = 푈 ⊆ R푛 and 푁 = 푉 ⊆ R푚 open. Let the local coordinates be 푥푖 and 푦푗 respectively. For 훼 = d푦푗, 푣 = 휕 and local coodinate for 푓 is 푦 = 푦 (푥), have 휕푥푘 푗 푗

∗ 휕 (푓 (d푦푗))( ) 휕푥푘 휕 =d푦푗((푑푓) ) 휕푥푘 휕푦 휕 =d푦 (∑ ℓ ) 푗 휕푥 휕푦 ℓ 푘 ℓ 휕푦 = ∑ ℓ 훿 휕푥 푗ℓ ℓ 푘 휕푦 = 푗 휕푥푘

∗ so 푓 is determined by its images on d푦푗.

5.4 de Rham cohomology We can use exterior derivatives to form a sequence

푑0 푑1 푑2 Ω0(푀) −→Ω1(푀) −→Ω2(푀) −→… with 푑푛+1 ∘ 푑푛 = 0 for all 푛. In fact this is a sequence of R-vector space homomorphisms.

Definition (closed form, exact form). If 훼 ∈ Ω푟(푀) is such that d훼 = 0 then 훼 is a closed form. If 훼 = d훽 for some 훽 ∈ Ω푟−1(푀) then 훼 is an exact form.

Clearly exact forms are closed but the converse is not true in general. Thus it makes sense to consider

Definition (de Rham cohomology). The quotient space

푟 ker 푑푟 퐻dR(푀) = , im 푑푟−1

sometimes also denoted 퐻푟(푀), is called the de Rham cohomology (or degree 푟) of 푀.

As 푓∗ commutes with d, 푓∗ is a well-defined map on 퐻푟(푀) and chain rule holds. Hence for 푀 diffeomorphic to 푁, we have

퐻푟(푀) ≅ 퐻푟(푁) for all 푟. The inverse, however, is not true.

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We can define ∞ ∗ 푟 퐻dR(푀) = ⨁ 퐻dR(푀). 푟=0 It is an easy exercise to check that wedge product induces a ring structure on ∗ 퐻dR(푀) (the key point is that (d훼)∧훽 = d(훼∧훽) whenever d훽 = 0). Moreover, 푟 푟 assume 푀 is connected, 퐻dR(푀) is finite-dimensional as Ω (푀) vanishes for 푟 푟 > dim 푀, so does 퐻dR(푀). For all smooth 푓 ∶ 푀 → 푁, the pullback 푓∗ induces a well-defined ring homomorphism 푓∗ ∶ 퐻∗(푁) → 퐻∗(푀). From previous work, have

Proposition 5.5. If 푀 is diffeomorphism to 푁 then

퐻∗(푀) ≅ 퐻∗(푁). Again the converse is false. Try to show that 퐻∗(R푃 3) ≅ 퐻∗(푆3). There are many types of (co)homologies on a space. The important result is

Theorem 5.6 (de Rham). Given a smooth manifold 푀,

푟 푟 퐻dR(푀) ≅ 퐻 (푀, R)

where 퐻푟(푀, R) is the singular cohomology of 푀 with coefficients in R. 푟 This tells us that 퐻dR(푀) recovers an invariant of the topological space underlying 푀, using differential 푟-forms. 0 If 푀 is connected then 퐻dR(푀) ≅ R, i.e. the constant functions on 푀. For more generally result, we need

Theorem 5.7 (Poincaré lemma). Let 푈 be the unit open ball in R푛. Then every closed 푘-form, where 푘 > 0, is exact.

Corollary 5.8. 0 푘 > 0 퐻푘(푈) = { R 푘 = 0

In fact, we can replace 푈 by R푛 and get the same result. Proof of Poincaré lemma. We give a sketch of proof. The key idea of is that we can invert d푘’s using integral operators 푘 푘−1 ℎ푘 ∶ Ω (푈) → Ω (푈) for 푘 > 0 such that ℎ푘+1 ∘ d푘 + d푘−1 ∘ ℎ푘 = idΩ푘(푈) . Explicitly, define ℎ (푎 푥 ∧ … ∧ 푥 )(푣 , … , 푣 ) 푘 d 푖푖 d 푖푘 1 푘−1 1 푘 휕 = (∫ 푡푘−1푎(푡푥)푑푡) 푥 ∧ … ∧ 푥 (∑ 푥 , 푣 , … 푣 ). d 푖1 d 푖푘 푖 1 푘−1 0 푖=1 휕푥푖

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5.5 Basic integration on manifolds In this section assume 푀 is an oriented 푛-manifold, where the orientation is given by a choice of positively compatible charts {(휑훼, 푈훼)}. Let 휔 ∈ Ω푛(푀) and define its support to be

supp 휔 = {푝 ∈ 푀 ∶ 휔(푝) ≠ 0}.

Assume the closure supp 휔 is comapct. If supp 휔 ⊆ 푈훼 for some 훼 with local coordinates 푥푖 and 휔| = 푓(푥) 푥 ∧ … ∧ 푥 , 푈훼 d 1 d 푛 then define

∫ 휔 = ∫ 푓(푥)푑푥1 … 푑푥푛 푀 휑훼(푈훼) where RHS is the Riemann integral on R푛. To show this is well-defined, suppose supp 휔 ⊆ 푈훽 for some 훽 with coordinates 푦푗, and

휔| = ℎ(푦) 푦 ∧ … ∧ 푦 . 푈훼∩푈훽 d 1 d 푛

Then

퐷푦 ∫ ℎ(푦)푑푦 = ∫ ℎ(푦(푥))∣ det ∣푑푥 = ∫ 푓(푥)푑푥 퐽 퐷푥 퐼 퐼 휑훽(푈훽) 휑훼(푈훼) 휑훼(푈훼)

To define the intergral of a differential form that is not supported on asingle chart, we use partition of unity.

Definition (integration on manifold). Suppose {휌푖} is a partition of unity subordinate to the oriented cover. Then define

∫ 휔 = ∑ ∫ 휌푖휔. 푀 푖 푈푖

Some basic properties of the integral whose proofs will be left as exercises: 1. it is linear in 휔, 2. it is additive over disjoint coordinate neighbourhoods,

3. it is independent of choice of partition of unity (hint: if {휌푖} and {푗 ̃휌 } are two partitions of unity then show {휌푖 푗̃휌 } is another).

Theorem 5.9 (Stoke’s theorem for manifolds without boundary). If 휂 ∈ Ω푛−1(푀) is compactly supported then

∫ d휂 = 0. 푀

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푁 Proof. Let {푈푖}푖=1 be positively compatible coordinate neighbourhoods cover- ing supp 휂 and let choose charts on 푈0 = 푀 \ supp 휂 so that it is positively 푁 compatible with 푈푖 for 1 ≤ 푖 ≤ 푁. Then {푈푖}푖=0 define an orientation on 푀. 푁 Let {휌푖} be a partition of unity subordinate to {푈푖}푖=0 and let

푁 d휂 = ∑ d(휌푖휂) 푖=0 be an 푛-form. Suffices to prove that for all 푖,

∫ d(휌푖휂) = 0. 푀

Fix 푖 and let 푥푘 be coordiantes on 푈푖. wlog

휌푖휂 = ℎd푥1 ∧ … ∧ d푥푛 so 휕ℎ d(휌푖휂) = d푥2 ∧ … ∧ d푥푛. 휕푥1 As supp ℎ is bounded, say by 푅, have

∫ 푑(휌푖휂) 푛 R 푅 휕ℎ = ∫ (∫ d푥1)푑푥2 … 푑푥푛 푛−1 휕푥 R −푅 1

= ∫ ⏟⏟⏟⏟⏟⏟⏟ℎ(푅, 푥2, … , 푥푛) − ℎ(−푅,⏟⏟⏟⏟⏟⏟⏟ 푥2, … , 푥푛) 푑푥2 … 푑푥푛 푛−1 R =0 =0 =0

Corollary 5.10 (integration by parts). Assume one of the forms 훼, 훽 is compactly supported on 푀 and deg 훼 + deg 훽 + 1 = dim 푀. Then

∫ 훼 ∧ d훽 = (−1)1+deg 훼 ∫ (d훼) ∧ 훽. 푀 푀

Proof. Apply Stoke’s theorem to 휂 = 훼 ∧ 훽.

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6 Vector bundles

Definition (submersion). A smooth map 푓 ∶ 푀 → 푁 is a submersion if 푑푓푥 ∶ 푇푥푀 → 푇푓(푥)푁 is surjective for all 푥 ∈ 푀.

Definition (vector bundle). A vector bundle over a manifold 퐵 is a smooth submersion 휋 ∶ 퐸 → 퐵 of a manifold 퐸 onto 퐵 satisfying

−1 1. there exists a vector space 푉 such that for all 푝 ∈ 퐵, 퐸푝 ∶= 휋 (푝) is a vector space isomorphic to 푉;

2. local trivailisation: for every 푝 ∈ 퐵, there exists open neighbourhood 푈 of 푝 and a diffeomorphism Φ푈 such that the diagram

Φ 휋−1(푈) 푈 푈 × 푉 휋

푈

commutes, where the map 푈 ×푉 → 푈 is projection to first coordinate;

3. Φ푝 ∶ 퐸푝 → {푝} × 푉 is an isomorphism of vector spaces for all 푝 ∈ 푈. 퐵 is called the base of the bundle and 퐸 is called the total space. 푉 is called typical bundle and dim 푉 is the rank of the vector bundle. 휋 is the bundle projection. Φ푈 is a local trivialisation and 푈 is a trivialising neighbourhood.

Vector bundles of rank 1 are also called line bundles.

Definition (section). A section of a vector bundle 퐸 is a smooth map 푠 ∶ 퐵 → 퐸 such that 휋 ∘ 푠 = id퐵. A local section 푠 ∶ 푁 → 퐸 where 푁 ⊆ 퐵 is a smooth map such that 휋 ∘ 푠 = id푁.

Example. 1. For any manifold 푉 and vector space 푉, take 퐸 = 퐵 × 푉 and 휋 to be the projection onto first coordinate. This is known asa trivial bundle or product bundle. Sections of this bundle are 퐶∞(퐵, 푉 ), i.e. vector valued functions on 퐵. 2. (Co)tangent bundles 푇 푀, 푇 ∗푀 are real vector bundles of rank dim 푀. Sections are 푉 (푀) and Ω1(푀) respectively. More generally, Λ푟푇 ∗푀 is 푛 푟 a real bundle of rank (푟) and sections are differential 푟-forms Ω (푀). If 푟 = 0 then the bundle is trivial.

3. Tautological vector bundle over R푃 푛, C푃 푛 and more generally Grassman- nians: set 퐵 = C푃 푛 and 퐸 to be the disjoint union of all complex lines in C푛+1 through 0 and 휋 maps 푧 ∈ 퐸 to the complex line containing 푧 in C푃 푛. We’ll check that this is a well-defined line bundle.

31 6 Vector bundles

6.1 Structure group and transition functions

Let (푈훼, 휑훼) and (푈훽, 휑훽) be two local trivialising neighbourhoods with 푈훼 ∩ 푈훽 ≠ ∅. Then −1 휑훽 ∘ 휑훼 (푏, 푣) = (푏, 휓훽훼(푏)(푣)) for all (푏, 푣) ∈ (푈훼 ∩ 푈훽) × 푉, where 휓훽훼 ∶ 푈훼 ∩ 푈훽 → GL(푉 ) is smooth. We have

휓훼훼 = id푉

휓훼훽 ⋅ 휓훽훼 = id푉

휓훼훽 ⋅ 휓훽훾 ⋅ 휓훾훼 = id푉 which are called the cocycle conditions. 휓훽훼 are called the transition functions of the vector bundle 퐸.

∗ Example. When 퐸 = 푇 푀 or 푇 푀, 휓훽훼 are given by the Jacobian matrices of a change of local coordinates (they are daul to each other).

Proposition 6.1. The data of 퐵 and {(푈훼, 휓훽훼)} with 퐵 = ⋃ 푈훼 deter- mines a vector bundle 퐸 uniquely up to isomorphism (which we will define formally later).

Proof. The proof is called Steenrod construction. Define

퐸 ∶= (∐ 푈훼 × 푉) / ∼ 훼 where the equivalence relation is the one generated by (푏, 푣) ∼ (푏, 휓훽훼(푏)(푣)) for all 푏 ∈ 푈훼 for all 훼, 훽. 퐸 is a manifold as ∼ is a diffeomorphism. wlog 푈훼 ⊆ 퐵 are coordinate neighbourhoods with charts 휑훼. Then Φ훼 = (휑훼, id) are charts for 퐸. Finally, 휋 ∶ 퐸 → 퐵, (푏, 푣) ↦ 푏 is a smooth submersion.

Definition (퐺-structure). Let 퐸 be a real vector bundle over 퐵. Let 퐺 ≤ GL(푘, R). A collection of charts {(푈훼, 휑훼)} covering 퐵 such that the corresponding 휓훽훼(푏) ∈ 퐺 for all 훼, 훽 is called a 퐺-structure. If 퐸 = 푇 푀, we say 퐸 is a 퐺-structure on 퐵.

Example. 1. If 퐺 is the trivial subgroup then a 퐺-structure is a global trivialisation over 퐵, i.e. Φ ∶ 퐸 → 퐵 × R푘 is a local trivilisation on 푈 = 퐵.

2. If 퐺 = GL+(푘, R), the linear isomorphisms with positive determinant, then a 퐺-structure is an orientation of 퐸. For example, if 퐸 = 푇 푀 then this gives an orientation of 푀. 3. If 퐺 = 푂(푘) (or 퐺 = 푈(푘) in case of complex bundle) then it determines a choice of invariantly defined inner product on fibres of 퐸 smoothly varying with the fibre. Φ훼 are then called orthogonal (unitary respectively) trivial- Φ ∶ 휋−1(푈 ) → 푈 × 푛 isations. In this case given a trivialisation 푈훼 훼 훼 R and

32 6 Vector bundles

푏 ∈ 푈 Φ | ∶ 휋−1(푏) → {푏} × 푛 훼, the linear isomorphism 푈훼 푏 R is moreover a linear isometry.

If we set instead 퐺 = 푆푂(푘) = 푂(푘) ∩ GL+(푘, R) we get an orthogonal trivialisation together with an orientation. 4. Suppose 퐸 is a real vector bundle of rank 푘 = 2푛 and take

퐺 = GL(푛, C) ⊆ GL(2푛, R), 푝 ∈ 퐵 퐽 ∈ (퐸 ) 퐽 2 = − then for all , there exists 푝 GL 푝 such that id퐸푝 which depends smoothly on 푝. In other words, (퐸푝, 퐽푝) is isomorphic to a complex vector space of dimension 푛. This 퐺-structure makes 퐸 into a complex vector bundle. Note that this is not the same as complex manifold, which has a stronger requirement. When 퐸 = 푇 푀 then a GL(푛, C)-structure on 퐸 is called an almost complex structure. Generally, if 퐺 ≤ GL(푉 ) preservse some “linear algebra objects” on 푉 then 퐺-structure on a vector bunldes means there exists a well-defined family of such objects depend smoothly on the point in the base.

6.2 Principal bundles

Let 퐺 be a Lie group, 1퐺 the identity element of 퐺. Let 푃 , 퐵 be manifolds.

Definition (smooth free right action). A smooth free right action is a right action of a Lie group on a manifold that is also free. Concretely, a smooth free right action of 퐺 on 푃 is a smooth map

푃 × 퐺 → 푃 (푝, ℎ) ↦ 푝ℎ

satisfying

1. free action: for all 푝 ∈ 푃, 푝ℎ = 푝 if and only if ℎ = 1퐺.

2. right action: for all 푝 ∈ 푃 , ℎ1, ℎ2 ∈ 퐺, (푝ℎ1)ℎ2 = 푝(ℎ1ℎ2). Note. The right action implies that for all ℎ ∈ 퐺, 푝 ↦ 푝ℎ is a diffeomorphism of 푃 to itself.

Definition (principal bundle). A principal 퐺-bundle of 푃 over 퐵 is a smooth submersion 휋 ∶ 푃 → 퐵 with a smooth free right action of 퐺 on 푃 such that the set of orbits is bijective with 퐵 “naturally”: for all 푏 ∈ 퐵, there exists an open neighbourhood 푈 of 푏 with a diffeomorphism Φ푈 such that

Φ 휋−1(푈) 푈 푈 × 퐺

휋

푈

commutes, where 푈 × 퐺 → 푈 is projection to first coordinate. Moreover Φ푈

33 6 Vector bundles

commutes with 퐺-action, i.e.

Φ푈(푝ℎ) = (푏, 푔ℎ)

whenever 푏 = 휋(푝) ∈ 푈, Φ푈(푝) = (푏, 푔). 푃 is the total space, 퐵 is the base and 퐺 is the fibre. Note. Principal 퐺-bundle is not the same as a locally trivial bundle with fibres being a Lie group. It also incorporates information about group action. Similar to vector bundles, given charts on two trivialising neighbourhoods, have −1 휑훽 ∘ 휑훼 (푏, 푔) = (푏, 휓훽훼(푏, 푔)) where 휓훽훼(푏, ⋅) ∶ 퐺 → 퐺. The 퐺-equivariant condition then implies that 휓 (푏, 푔) = 휓 (푏)푔 = 퐿 푔 훽훼 훽훼 휓훽훼(푏) where 퐿ℎ ∶ 퐺 → 퐺 is left multiplication by an element ℎ and 휓훽훼(푏) = 휓훽훼(푏, 1퐺). As before, the transition functions 휓훽훼 ∶ 푈훼 ∩ 푈훼 → 퐺 are smooth maps satisfying the cocycle conditions. We can also recover a principal 퐺-bundle from the transition functions, up to 퐺-bundle isomorphism:

Theorem 6.2. Given {(푈훼, 휓훽훼)} such that 퐵 = ⋃ 푈훼 and 휓훽훼’s satisfy the cocycle conditions, let

푃 ∶= (∐ 푈훼 × 퐺) / ∼ 훼

′ ′ ′ ′ where (푏, ℎ) ∼ (푏 , ℎ ) if and only if 푏 = 푏 , ℎ = 휓훽훼(푏)ℎ. Then 푃 is a principal 퐺-bundle over 퐵.

Proof. Same as before. Remark. The orbit space can be made into a manifold with an additional assumption: if the action of 퐺 is also proper, i.e. the map 퐺 × 푃 → 푃 × 푃 (푔, 푝) ↦ (푝푔, 푝) is a proper map (preimage of compact set is compact) then can show 푃 /퐺 is a smooth manifold. Then may define 퐵 = 푃 /퐺. The consequence of the above theory is that we can start from a vector bundle 퐸 with a 퐺-structure, and use {휓훽훼} to obtain a principal 퐺-bundle 푃 with the same transition functions. Conversely, given a principal 퐺-bundle 푃, we say 퐺 is associated to 푃 via the action of 퐺 on R푛 where 푛 is the rank of 퐸, i.e. this is a representation 퐺 → GL(푛, R), an injective homomorphism. In the case of principal 퐺-bundle, 퐺 acts on itself by left translation. Example. Let 퐸 = 푇 푀 then 푃 is a frame bundle, 퐺 = GL(푛, R). For all 푝 ∈ 푃, it is a basis of 푇휋(푝)푀. If in addition 푇 푀 has 푂(푛)-structure then obtain 푃 the orthonormal frames bundle.

34 6 Vector bundles

6.3 Hopf bundle A Hopf bundle is an example of a tautological rank 1 complex vector bundle 1 1 2 over C푃 . Recall that the fibre over 푧1 ∶ 푧2 ∈ C푃 is the line C(푧1, 푧2) ⊆ C . We shall work out the transition functions to prove this vector bundle is well-defined. Let 1 푈푖 = {푧푖 ≠ 0} ⊆ C푃 and let 푧 = 푧2 be a local coordinate on 푈 , 휁 = 푧1 on 푈 . Every point in fibre 푧1 1 푧2 2 over 1 ∶ 푧 ∈ 푈1 can be written as (푤, 푤푧) where 푤 ∈ C, and respectively as (푤휁, 푤) over 휁 ∶ 1 ∈ 푈2. The local trivialisations over 푈푖 are −1 휑1 ∶ 휋 (푈1) → 푈1 × C (푤, 푤푧) ↦ (1 ∶ 푧, 푤√1 + |푧|2) −1 휑2 ∶ 휋 (푈2) → 푈2 × C (푤휁, 푤) ↦ (휁 ∶ 1, 푤√|휁|2 + 1) then ̃푤 ̃푤 휑−1(1 ∶ 푧, ̃푤) = ( , 푧) 1 √1 + |푧|2 √1 + |푧|2 if 푧 ≠ 0 then

̃푤 ̃푤 휑 ∘ 휑−1(1 ∶ 푧, ̃푤) = 휑 ( 휁, ) 2 1 2 휁√1 + 1/|휁|2 휁√1 + 1/|휁|2 |휁| = (휁 ∶ 1, ̃푤) 휁 푧 = (1 ∶ 푧, ̃푤) |푧| by noting that 휁푧 = 1. Thus the transition functions are 푧 휓 (1 ∶ 푧) = 2,1 |푧| |푧| 휁 휓 (휁 ∶ 1) = = 1,2 푧 |휁| Observe

∗ 휓2,1 ∶ 푈1 ∩ 푈2 → 푈(1) = {푎 ∈ C ∶ |푎| = 1} ⊆ C = GL(C) so the Hopf-bundle is well-defined and admits a 푈(1)-structure. Thus it has an invariantly defined norm on the fibres, induced from standard Euclidean norm 2 −1 on C ⊆ C via 휑1, 휑2. More explicitly, on 휋 (푈1), ‖(푤, 푤푧)‖ = |푤|√1 + |푧2|.

−1 Leave as an exercise to write out the expression for 휋 (푈2), and verify that it gives the same norm. The associated principal 푈(1)-bundle 푃, also known as Hopf bundle, is the bundle of unit vectors. Thus

2 2 2 3 푃 ≅ {(푤1, 푤2) ∈ C ∶ |푤1| + |푤2| = 1} = 푆

35 6 Vector bundles and

휋 ∶ 푆3 → C푃 1

(푤1, 푤2) ↦ 푤1 ∶ 푤2

But recall from example sheet 1 Q5 that C푃 1 ≅ 푆2 so 휋 ∶ 푆3 → 푆2. Together with some topological argument, we see that the Hopf bundle cound not be a product bundle 푆3 ≅ 푆2 × 푆1. See example sheet 2 Q5.

Definition (local section). A local section of a principal bundle 휋 ∶ 푃 → 퐵 is a smooth map 푠 ∶ 푊 → 푃, where 푊 ⊆ 퐵 open, such that 휋 ∘ 푠 = id푤. There is nothing new in the dfinition here — we have seen local and global sections for vector bundles before. However, note that although every vector bundle admits a global section, namely the zero section, it is not true for prin- cipal bundles. We cannot choose the “identity element section” as there is no distinguished point in a fibre. In fact, a principal bundle admits a global section if and only if it is trivial.

6.4 Pullback of a vector bundle or a principal bundle

Definition (pullback of a vector bundle). Let 휋 ∶ 퐸 → 퐵 be a vector bundle and 푓 ∶ 푀 → 퐵 be a smooth map. The pullback 푓∗퐸 of 퐸 is a vector bundle over 푀 with same fibre 푉 as 퐸 and such that there exists smooth 퐹 making the diagram 푓∗퐸 퐹 퐸

̃휋 휋 푓 푀 퐵 ∗ commutes, and the restriction 퐹 ∶ (푓 퐸)푝 → 퐸푓(푝) to fibres is an isomor- phism of vector spaces for all 푝 ∈ 푀.

It follows that for all local trivialisations of 퐸 over 푈, we must have

Φ 휋−1(푈) 푈 푈 × 푉

퐹 푓×id푉 Φ̃ ̃휋−1(푓−1(푈)) 푈 푓−1(푈) × 푉

̃ It is left as an exercise to check that Φ푈 is a diffeomorphism. Example. 1. If 퐵 = ∗ then 퐸 ≅ 푉. Have Φ̃ ∶ 푓∗퐸 → 푀 × 푉 푥 ↦ ( ̃휋(푥), 퐹 (푥))

as a (global) trivialisation of 푓∗퐸 over 푀.

36 6 Vector bundles

2. As a variant, let 푀 = 퐵 × 푋 with 푓 ∶ 퐵 × 푋 → 퐵 the projection to first coordinate. Then 푓∗퐸 is “trivialised in the 푋 direction”, i.e. 푓∗퐸 ≅ 퐸 ×푋 and ̃휋= 휋 × id푋. 3. Suppose 푀 = ∗. Then 푓∗퐸 is just a copy of the fibre 푉 and 퐹 ∶ 푉 ↪ 퐸 is an embedding as the fibre over 푓(푀) ∈ 퐵. In general, 푓∗퐸 may be determined by pulling back the transitions

∗ −1 푓 휓훽훼 = 휓훽훼 ∘ 푓 ∶ 푓 (푈훼 ∩ 푈훽) → GL(푉 ). This gives an equivalent but more formal way to define pullbacks of bundles. The above discussion applies equally well to pullbacks of principal 퐺-bundles 푓∗푃, with obvious change of notations.

6.5 Bundle morphism

Definition (vector bundle morphism). Let 휋 ∶ 퐸 → 퐵, 휋′ ∶ 퐸′ → 퐵′ be two vector bundles and 푓 ∶ 퐵 → 퐵′ a smooth map. A smooth map 퐹 ∶ 퐸 → 퐸′ is a vector bundle morphism (covering 푓) if the diagram

퐸 퐹 퐸′

휋 휋′ 푓 퐵 퐵′

푝 ∈ 퐵 퐹 = 퐹 | ∶ 퐸 → 퐸′ commutes and for all , the restriction 푝 퐸푝 푝 푓(푝) is a linear map.

It follows that if Φ, Φ′ are local trivialisations of 퐸 and 퐸′ respectively over 푈 ⊆ 퐵, 푈 ′ ⊆ 퐵′ with 푓(푈) ⊆ 푈 ′, then the local expression 퐹̂ = Φ′ ∘ 퐹 ∘ Φ−1 is given by 퐹̂ (푏, 푣) = (푓(푣), ℎ(푏)(푣)) for all 푏 ∈ 푈 and ℎ ∶ 푈 → 퐿(푉 , 푉 ′) is a smooth map. Example.

1. Suppose 휑 ∶ 푀 → 푁 is a smooth map then 푑휑 ∶ 푇 푀 → 푇 푁 given by 푑휑푝 for all 푝 ∈ 푀 is a bundle morphism. 2. Given 푓 ∶ 푀 → 퐵 and 휋 ∶ 퐸 → 퐵 a vector bundle, the pullback 퐹 ∶ 푓∗퐸 → 퐸 is a bundle morphism.

′ 3. Let 퐵 = 퐵 and 푓 ∶ 퐵 → 퐵 a diffeomorphism. If for all 푝 ∈ 퐵, 퐹푝 ∶ 퐸푝 → 퐸푓(푝) is a linear isomorphism then 퐹 is an vector bundle isomorphism. ′ ′ Furthermore if 푓 = id퐵, 퐸 = 퐸, 휋 = 휋 then we have the commutative diagram 퐸 퐹 퐸 휋 휋 퐵 In this case 퐹 is called an automorphism of 퐸 and denote it by 퐹 ∈ Aut(퐸).

37 6 Vector bundles

4. For example if 퐸 ≅ 푉 × 푉 then Aut(퐸) = 퐶∞(푉 , GL(푉 )). If 퐸 has a 퐺-structure then Aut퐺(퐸) ⊆ Aut(퐸) is well-defined. Locally

퐹̂ (푏, 푣) = (푣, ℎ(푏)(푣))

where ℎ(푣) ∈ 퐺 ⊆ GL(푉 ), whenever 퐹̂ is with respect to local trivial- isations Φ from this 퐺-structure. In mathematical physics, often write Aut퐺(퐸) = 풢, the group of gauge transformations. For example, 퐺 = 푈(1), 푆푈(2), 푆푂(3), 푆푈(3).

38 7 Connections

7 Connections

Let 휋 ∶ 퐸 → 퐵 be a vector bundle and 푠 ∶ 퐵 → 퐸 is a section. Locally ̂푠∶ 푈 → 푈 × 푉 푏 ↦ (푏, 푠(푏)) which is really, just a map to 푉 so 푑푝 ̂푠 ∶ 푇푏푈 → 푇푠(푏)(푉 ) ≅ 푉. On the other hand, the ordinary differential of 푠 is a map 푑푠푏 ∶ 푇푏퐵 → 푇(푏,푠(푏))퐸. We would like 푇푠(푏)퐸푏 ≅ 퐸푏 for some subspace of 푇푠(푏)퐸. How to define differential of a section of a vector bundle so that itlooks reasonable like a gradient Some conventions throughout: let dim 퐵 = 푛 and 푈 ⊆ 퐵 a coordinate neighbourhood that is simultaneously a trivialising neighbourhood for 퐸, with 푘 푗 푚 푚 coodinates 푥 where 푘 = 1, … , 푛. Let 푎 = (푎 )푗=1 ∈ R be coordinates on the fibres on 퐸. We use 푖, 푗 = 1, … , 푚 and 푘 = 1, … , 푛 and use summation convection for Roman indices (but not Greek indices). −1 푚 Let Φ푈 ∶ 휋 (푈) → 푈 × R be a local trivialisation. Let 푇푝 be the span of 휕 휕 −1 휕푥푘 , 휕푎푗 . For 푝 ∈ 퐸, 휋(푝) = 푏, Let 휋 (푏) = 퐸푏 ⊆ 퐸 is a submanifold.

푇푝퐸푏 = ker(푑휋푝) 휕 which is the span of 휕푎푗 .

Definition. The vertical subspace at 푝 ∈ 퐸 is

푇 푣푝퐸 = ker(푑휋푝).

A subspace 푆푝 ⊆ 푇푝퐸 is a horizontal subspace if

푆푝 ∩ 푇 푣푝퐸 = 0

푆푝 ⊕ 푇 푣푝퐸 = 푇푝퐸

Then dim 푆푝 = dim 퐵. How to choose a horizontal subspace? All 푛 dimensional subspace of 푇푝퐸 ≅ 푚+푛 ⋂푚 휃푖 휃1, … , 휃푚 ∈ ( 푚+푛)∗ ≅ 푇 ∗퐸 R is 푖=1 ker for some linearly independent R 푝 where 푖 푖 푘 푖 휃 = 푓푘d푥 + 푔푗d푎푗 푖 푖 푘 휕 푗 휕 where 푓푘, 푔푗 ∈ R. A subspace 퐶 = 퐵 휕푥푘 + 퐶 휕푎푗 ∈ 푇푝퐸 is vertical if and only if 퐵푘 = 0 for all 푘. Thus must have 휕 휃푖 (퐶 푝 푗 휕푎푗 푖 푗 푗 not all 0 for 푖 = 1, … , 푛, i.e. 푔푗퐶 = 0 for all 푖, which implies 퐶 = 0 for all 푗. 푖 푖 Hence the 푚 × 푚 matrix (푔푗) is invertible. Let its inverse be (ℎ푗). Put 푖̃ 푖 푗 푗 푖 푘 휃푝 ∶= ℎ푗휃푝 = d푎 + 푒푘d푥 which determines the same subspace 푆푝 ⊆ 푇푝퐸, where 푖 푖 푖 ∞ 푛 푒푘 = 푒푘(푝) = 푒푘(푥, 푎) ∈ 퐶 (푈 × R ).

39 7 Connections

Proposition 7.1. Every given field of horizontal subspace 푆푝, 푝 ∈ 퐸 is locally expressed as 푚 푖 푆푝 = ⋂ ker 휃푝 푖=1 where 푖 푖 푖 푘 휃푝 = d푎 + 푒푘(푥, 푎)d푥 푖 푚 for some unique functions 푒푘 on 푈 × R . 푖 We say 푆 = 푆푝 is smooth if all 푒푘 are smooth.

Definition (connection). 푆 = 푆푝 a field of horizontal subspaces isa con- 푖 nection of 퐸 if all 푒푘’s are linear in 푎. Write

푖 푖 푗 푒푘(푥, 푎) = Γ푗푘(푥)푎

푖 ∞ where Γ푗푘 ∈ 퐶 (푈) are the coefficients of this connection in a local trivial- isation.

푖 푖 푖 푗 푘 푖 푖 푗 휃 = d푎 + Γ푗푘(푥)푎 d푥 = d푎 + 퐴푗푎 푖 1 where 퐴푗 ∈ Ω (푈). 푖 ′ The transformation law for 퐴푗: if 푈 is another trivialisation neighbourhood ′ 푖′ with 푈 ∩ 푈 ≠ ∅ and Ψ = (Ψ푖 )푚×푚 is the transition function from Φ푈 to Φ푈′ , −1 푖 and Ψ = (Ψ푖′ )푚×푚 the transition function fro Φ푈′ to Φ푈. Note that

푖 푖′ 푖 Ψ푖′ Ψ푗 = 훿푗.

푖′ 푖′ 푖 Suppose 푎 = Ψ푖 푎 , then

푖′ 푖′ 푖 푖′ 푖 d푎 = (푑Ψ푖 )푎 + Ψ푖 푑푎 .

Suppose

푖′ 푖′ 푖′ 푗′ 휃 = 푑푎 + 퐴푗′ 푎 푖 푖 푖 푗 휃 = 푑푎 + 퐴푗푎 then rewrite 푖′ 푖′ 푖 푖′ 푖 푖′ 푗′ 휃 = (푑Ψ푖 )푎 + Ψ푖 푑푎 + 퐴푗′ 푎 Compare the coefficient of 푑푎푖, must have

푖 푖 푖′ 휃 = Ψ푖′ 휃 so 푖 푖 푖 푖′ 푖 푖′ 푗′ 푗 휃 = 푑푎 + (Ψ푖′ 푑Ψ푗 + Ψ푖′ 퐴푗′ Ψ푗 )푎 so the transformation law is

푖 푖 푖′ 푗′ 푖 푖′ 퐴푗 = Ψ푖′ 퐴푗′ Ψ푗 + Ψ푗′ 푑Ψ푗 .

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Φ 푖 Φ′ ΨΦ 푖′ Use 퐴 to denote the matrix (퐴푗), have 퐴 = 퐴 = (퐴푗′ ). Then the trans- formation law is 퐴ΨΦ = Ψ퐴ΦΨ−1 − (푑Ψ)Ψ−1. Noting that (푑Ψ)Ψ−1 + Ψ푑(Ψ−1) = 0 so equally 퐴Ψ = Ψ−1퐴ΨΦΨ + (푑Ψ)Ψ−1.

푖 Theorem 7.2. Every choice of local matrices (퐴푗) of 1-forms satifying * assigned to local trivialisations defines a connection on 퐸.

Recall the transformation law for the connection matrices

퐴ΨΦ = Ψ퐴Φ휓−1 − (푑Ψ)Ψ−1 where Ψ is the transition function from local trivialisation Φ to Φ′ with Φ = ′ −1 Φ 푖 Φ ∘ Φ . Write 퐴 = (퐴푗)푖,푗=1,…,푚 where 푚 is rank of 퐸,

푖 푖 푘 1 퐴푗 = Γ푗푘(푥)d푥 ∈ Ω (푈).

ΨΦ 푖′ 푖 Similarly 퐴 = (퐴푗′ ). The (퐴푗)’s do not patch together over trivialising neighbourhoods 푈 ⊆ 퐵 to give a global matrix-valued 1-form, essentialy because of the (푑Ψ)Ψ−1 correction term in the transformation law. −1 Let Φ훼,Φ훽 ∶ 휋 (푈) → 푈 × 푉 be two trivialising neighbourhoods over 푈. If 퐺훼 ∈ End(푉 ) considered in the Φ훼 trivialisation then with respect to Φ훽, 퐺훼 becomes 퐺훽 = Ψ훽훼퐺훼Ψ훼훽, note that there is no summation here! This defines a linear action of GL(푉 ) on End(푉 ). Thus we can apply the Steenrod construction to build a vector bundle End(퐸) using 푈 × End(푉 ) and the above formula for the transitions. This is the endomorphism bundle of 퐸. This is the bundle associated to 퐸. Remark. We may also define subbundle GL(퐸) ⊆ End(푉 ) with typical fibre GL(푉 ) ⊆ End(푉 ). GL(퐸) is not a principal bundle. For example it always has a global section

퐵 → GL(퐸)

푏 ↦ id푉

Sections of GL(퐸) are precisely Aut(퐸). Coincidentally this also provides a practical counterexample to the statement that “every vector bundle with tran- sition functions in a group is a principal bundle”. Similarly to End(퐸) we may build a vector bundle with typical fibre

Λ푟(R푛)∗ ⊗ 푉 ≅ {푓 ∶ R⏟⏟⏟⏟⏟푛 × ⋯ × R푛 → 푉 multilinear and antisymmetric} 푟 times

(where the isomorphism comes from the fact in linear algebra that (R푛)∗ ⊗ 푉 ≅ {linear maps R푛 → 푉 } and so on), where 푛 = dim 퐵. Use the action induced

41 7 Connections

vector bundle typical fibre transition functions 퐸 푉 푣 ↦ Ψ훽훼푣 End(퐸) End(푉 ) 퐺 ↦ Ψ훽훼퐺Ψ훼훽 푇 ∗퐵 ⊗ 퐸 퐿( 푛, 푉 ) ≅ ( 푛)∗ ⊗ 푉 푣 푥푘 ↦ (Φ 푣 ) 휕푥푘 푥푘′ R R 푘d 훽훼 푘 휕푥푘′ d Λ푟(푇 ∗퐵) ⊗ 퐸 Λ푟( 푛)∗ ⊗ 푉 푣 푥퐾 ↦ (Φ 푣 ) 퐷푥퐾 푥퐾′ R 퐾d 훽훼 퐾 퐷푥푘′ d 푇 ∗퐵 ⊗ 퐸 ( 푛)∗ ⊗ 푉 퐺 푥푘 ↦ (Ψ 퐺 Ψ ) 휕푥푘 푥푘′ End R End 푘d 훽훼 푘 훼훽 휕푥푘′ d Λ푟푇 ∗퐵 ⊗ End 퐸 Λ푟(R푛)∗ ⊗ End 푉

by the Jacobi matrices for local coordinate transformation on 퐵 and use Ψ훼훽’s on the 푉 factor. whwere we multiply vectors and matrices in Λ푟푉 ⊗퐸 and Λ푟푉 ⊗End 푉 using the wedge product of forms. (The good news is we mostly use 푟 = 1, 2) Connections are not sections of 푇 ∗퐵 ⊗ End 퐸, but if 퐴 and 퐴̃ are two con- nections then 퐴 − 퐴̃ is a valid section of 푇 ∗퐵 ⊗ End 퐸. Thus the space of all connections 퐴 on a given vector bundle 퐸 is an affine space modeled on section ∗ 1 of 푇 퐵 ⊗ End 퐸, written Ω퐵(End 퐸). (this means there is no canonical zero section).

푟 푟 Notation. Use Γ(퐸) to denote all sections of 퐸 and Ω퐵(퐸), Ω퐵(End 퐸) to 푟 ∗ 푟 ∗ 0 denote sections of Λ 푇 퐵⊗퐸 and Λ 푇 퐵⊗End 퐸. In particular Ω퐵(퐸) = Γ(퐸). Now we are ready to define calculus on vector bundles.

Definition (covariant derivative). A covariant derivative on a real vector 퐸 1 bundle 퐸 over 퐵 is a R-linear map ∇ ∶ Γ(퐸) → Ω퐵(퐸) satisfying the Leibniz rule ∇퐸(푓푠) = d푓 ⊗ 푠 + 푓∇퐸푠 for all 푠 ∈ Γ(퐸), 푓 ∈ 퐶∞(퐵).

Example. Let 퐴 be a connection on 퐸. Have in any local trivialisation 퐴 = 푖 푖 1 (퐴푗) where 퐴푗 ∈ Ω (푈). Define an map d퐴 which is locally given by

푖 푖 푗 d퐴푠|푈 = (d푠 + 퐴푠)|푈 = (d푠 + 퐴푗푠 )푖=1,…,푚 where 푚 is rank of 퐸. To check this is well-defined, if 푈 is also a coordinate neighbourhood in 퐵 then

휕푠푖 (d 푠)푖 = ( + Γ푖 (푥)푠푗) d푥푘. 퐴 휕푥푘 푗푘

Let Φ′ be another local trivialisation over 푈. Then

푠 = Ψ푠′ 퐴 = Ψ퐴′Ψ−1 − (푑Ψ)Ψ−1

42 7 Connections so

d퐴(푠) = d푠 + 퐴푠 = d(Ψ푠′) + (Ψ퐴Ψ−1 − (푑Ψ)Ψ−1)Ψ푠′ = Ψd푠′ + (푑Ψ)푠′ + Ψ퐴′푠′ − (푑Ψ)푠′ = Ψ(d푠′ + 퐴′푠′) ′ = Ψ(d퐴푠)

1 which is the correction transformation law for Ω퐵(퐸).

퐸 퐸 Theorem 7.3. Every covariant derivative ∇ arises as ∇ = 푑퐴 for some connection 퐴 on 퐸. Recall that 푑퐴푆 = 푑푠 + 퐴푆 where RHS only makes sense with a choice of local trivialisation!

퐸 Proof. Claim every ∇ is a local operator. i.e. for all 푈 ⊆ 퐵 open, if 푠1|푈 = 푠2|푈 퐸 퐸 then ∇ 푠1|푈 = ∇ 푠2|푈. (remark by writer of note: essentially partition of unity subject to {푈, 퐵 \ 푈0) Indeed for all 푏 ∈ 푈 let

푏 ∈ 푈0 ⊆ 푈0 ⊆ 푈

∞ and let 훼 ∈ 퐶 (퐵) such that 0 ≤ 훼 ≤ 1, 훼|퐵\푈 = 0. Then

퐸 퐸 0 = ∇ (훼(푠1 − 푠2)) = 푑훼 ⊗ (푠1 − 푠2) + 훼∇ (푠1 − 푠2)

At 푏, 푑훼푏 = 0, 훼(푏) = 1 whence

퐸 퐸 (∇ 푠1)(푏) = (∇ 푠2)(푏)

1 푚 푖 so it suffices to work in trivialising neighbourhoods. Let 퐬 = (푠 , … , 푠 ) = 푠 퐞푖 푚 푖 ∞ where 퐞푖 is the standard basis of R and 푠 ∈ 퐶 (푈). Put

휕 푖 Γ푖 = [(∇퐸퐞 ) ] 푗푘 푗 휕푥푘

푖 where we assumed that 푈 is also a coordinate neighbourhood in 퐵. Γ푗푘 ∈ 퐶∞(푈). Thus

퐸 퐸 푖 푖 푗 푖 푘 ∇ 퐬 = ∇ (푠 퐞푖) = (푑푠 + 푠 Γ푗푘(푥)푑푥 ) ⊗ 퐞푖 = 푑퐴푠

푖 푖 푘 Then the previous example verifies that 퐴푗 = Γ푗푘푑푥 have the required trans- formation law.

Definition (covariantly constant). A (local) section 푠 ∶ 푈 → 퐸 is covari- antly constant with respect to connection 퐴 if 푑퐴푠 = 0.

This is geometer’s constant vector function.

43 7 Connections

Proposition 7.4. (푑퐴푠)(푥) = 0 if and only if (푑푠)푥(푇 푥퐵) ⊆ 푇푠(푥)퐸 is a horizontal subspace (푑푠푥(푇푥퐵) = 푆푠(푥) associated with 퐴.

Proof. Exercise.

푟 푟+1 We can extend 푑퐴 as a R-linear map 푑퐴 ∶ Ω퐵(퐸) → Ω퐵 (퐸) for 푟 = 0, 1, … by requiring deg 휎 푑퐴(휎 ∧ 휔) = (푑퐴휎) ∧ 휔 + (−1) 휎 ∧ 푑휔 푞 푝 for all 휎 ∈ Ω퐵(퐸), 휔 ∈ Ω퐵(퐵). Locally

퐼 퐼 퐼 퐼 퐼 푑퐴(푠퐼푑푥 ) = (푑퐴푠퐼) ∧ 푑푥 = 푑(푠푖푑푥 ) + (퐴 ∧ 푠 )푑푥 , i.e. 푑퐴휎 = 푑휎 + 퐴 ∧ 휎. Again, RHS only makes sense with a choice of local trivialisation. Can futher extend 푑퐴 to 푟 푟+1 푑퐴 ∶ Ω퐵(End 퐸) → Ω퐵 (End 퐸) via (푑퐴퐶)푠 = 푑퐴(퐶푠) − 퐶푑퐴푠 for all 퐶 ∈ Γ(End 퐸), 푠 ∈ Γ(퐸) (this is really just product rule rearranged). Also deg 휇 (푑퐴휇)휔휎 = 푑퐴(휇 ∧ 휎) − (−1) 휇 ∧ 푑퐴휎 푝 푟 for 휇 ∈ Ω퐵(End 퐸), 휎 ∈ Ω퐵(퐸). Finally

deg 휇1 푑퐴(휇1 ∧ 휇2) = (푑퐴휇 − 1) ∧ 휇2 + (−1) 휇1 ∧ 푑퐴휇

∗ for 휇1, 휇2 ∈ Ω퐵(End 퐸). 2 Example. For 휇 ∈ Ω퐵(End 퐸), the above theory implies in that in any local trivialisation, 푑퐴휇 = 푑휇 + 퐴 ∧ 휇 − 휇 ∧ 퐴.

7.1 Curvature

We may repeatedly apply covariant derivative 푑퐴 to get a chain

푑 푑 푑 푑 0 퐴 1 퐴 퐴 푛 퐴 Γ(퐸) = Ω퐵(퐸) −→Ω퐵(퐸) −→… −→Ω퐵(퐸) −→0.

Unfortunately in general 푑퐴 ∘ 푑퐴 ≠ 0 so we don’t have a cohomology. However 푟 in a local trivilisation, let 푠 ∈ Ω퐵(퐸), have

푑퐴푑퐴푠 = 푑(푑푠 + 퐴 ∧ 푠) + 퐴 ∧ (푑푠 + 퐴 ∧ 푠) =푑퐴∧푠−퐴∧푑푠+퐴∧푑푠+퐴∧퐴∧푠 = (푑퐴 + 퐴 ∧ 퐴) ∧ 푠 so 푑퐴푑퐴 is multiplication by a 2-form. The expression is a pointwise (i.e. alge- braic) expression: if 푠1(푏) = 푠2(푏) then

(푑퐴푑퐴푠1)푏 = (푑퐴푑퐴푠2)푏.

44 7 Connections

Note. One may wonder why 퐴 ∧ 퐴 does not vanish in the expression. In fact it is matrix valued:

푖 푖 푘 푝 ℓ 푖 푝 푘 ℓ (퐴 ∧ 퐴)푗 = Γ푝푘푑푥 ∧ Γ푗ℓ푑푥 = Γ푝푘Γ푗ℓ푑푥 ∧ 푑푥 ≠ 0 which is nonvanishing in general unless 퐸 has rank 1. In summary, matrix multiplicaiton is noncommutative.

Definition (curvature). Given a connection 퐴, the 2-form

2 퐹 (퐴) = 푑퐴 + 퐴 ∧ 퐴 ∈ Ω퐵(End 퐸)

is the curvature of 퐴.

Note that 퐹 (퐴) is well-defined (i.e. independent of local trivialisation) as 푑퐴푑퐴 is so. 휕Γ푖 Locally 퐹 퐴 = 퐹 (퐴)푖 푑푥푘∧푑푥ℓ where {퐹 (퐴)푖 } is a function of {Γ푖 , 푗푘 }. 푗,푘ℓ 푗,푘ℓ 푗푘 휕푥푒푙푙

Definition (flat). A connection is flat if 퐹 (퐴) = 0. A vector bundle 퐸 is flat if made a choice of flat connection.

Example. Consider the product bundle 퐸 = 퐵 × R푚. Then

∞ 푚 1 푚 푑퐴 = 푑 ∶ 퐶 (퐵, R ) → Ω (퐵) ⊗ R is a trivial product connection. It is flat. The converse is true only locally and only if connection is flat.

Theorem 7.5 (2nd Bianchi identity).

푑퐴퐹 (퐴) = 0

for every connection 퐴 on vector bundle 퐸.

Proof. Let 푠 ∈ Γ(퐸). Then

푑퐴(퐹 (퐴)푠) = 푑퐴(푑퐴푑퐴)푠 = (푑퐴푑퐴)푑퐴푠 = 퐹 (퐴) ∧ 푑퐴푠.

Bu LHS may be rewritten as

푑퐴(퐹 (퐴))푠 + 퐹 (퐴) ∧ 푑퐴푠.

Substitute and cancel to get the result.

45 8 Riemannian geometry

8 Riemannian geometry

Definition. A Riemannian metric 푔 on 푀 is a field of positive-definite symmetric bilinear forms

푔푝 ∶ 푇푝푀 × 푇푝푀 → R

defined for all 푝 ∈ 푀 and smooth in 푝.

Definition (Riemannian manifold). A Riemannian manifold is a manifold endowed with a Riemannian metric.

Remark. Smoothness in 푝 means that for all 푋, 푌 ∈ 푉 (푀) have 푔(푋, 푌 ) ∈ 퐶∞(푀). Equivalently, in each coordinate neighbourhood 푈 with coordinates 푥푖 have 휕 휕 푔 = 푔( , ) ∈ 퐶∞(푈). 푖푗 휕푥푖 휕푥푗 We thus obtain the local expression of 푔

푖 푗 푔푖푗푑푥 푑푥 .

푖 푗 Technically it should be 푔푖푗푑푥 ⊗푑푥 but we omit the tensor symbol. This means 푖 휕 푖 휕 that suppose 푋 = 푋 휕푥푖 , 푌 = 푌 휕푥푖 then

푖 푗 푔(푋, 푌 ) = 푔푖푗푋 푌 .

Formally 푔 ∈ Γ(푇 ∗푀 ⊗ 푇 ∗푀).

Example. This is compatible with the more classical definition of Riemannian metric in differential geometry of curves and surfaces. Given 푟 = 푟(푢, 푣) ∶ 2 3 R푢,푣 → R , the first fundamental form

퐸푑푢2 + 2퐹 푑푢푑푣 + 퐺푑푣2 is a Riemannian metric with 푔11 = 퐸, 푔12 = 푔21 = 퐹 , 푔22 = 퐺.

Theorem 8.1. Every manifold admits a Riemannian metric.

Proof. Apply example sheet 3 question 2 (every vector bundle can be given a smooth inner product) to 푇 푀. The definition of a pullback 퐹 ∗ for smooth 퐹 ∶ 푀 → 퐵 is valid for bilinear ∗ forms of 푇 푁. If 푔푁 is a metric on 푁 then 퐹 푔푁 is a symmetric, bilinear smooth and non-negative definite on fibres of 푇 푀. ∗ If also 푑퐹푥 are injective for all 푥 ∈ 푀 then 퐹 푔푁 is positive-definite on 푇 푀 and is a valid Riemannian metric. For example, when 푀 is a immersed submanifold or 퐹 is a diffeomorphism. This gives us an alternative way to prove the theorem by embedding in Euclidean space by Whitney and pullback the Euclidean metric.

46 8 Riemannian geometry

Definition (connection). A connection on a manifiold 푀 is a connection on 푇 푀. Recall that transition functions for 푇 푀 are given by the Jacobian 휕푥푖 휓 = (휓푖 ) = ( ) 푖′ 휕푥푖′

푖 The coefficients Γ푗푘 of a connection on 푀 are sometimes called Christoffel sym- bols. Recall the transformation law for connection

푘′ 푖′ ′ ′ 휕푥 휕휓 Γ푖 = Γ푖 휓푖 휓푗 + 휓푖 푗 푗푘 푗′푘′ 푖′ 푗 휕푥푘 푖′ 휕푥푘 When 퐸 = 푇 푀 this simplifies to

푖 푗′ 푘′ 푖 2 푖′ ′ 휕푥 휕푥 휕푥 휕푥 휕 푥 Γ푖 = Γ푖 + 푗푘 푗′푘′ 휕푥푖′ 휕푥푗 휕푥푘 휕푥푖′ 휕푥푗휕푥푘 푖 Note that in this case the object Γ푘푗 obtained by switching 푗 and 푘 makes sense as they live in the same space. It obeys the same transformation law so they also define a connection. 푖 푖 푖 The difference 푇푗푘 = Γ푗푘 − Γ푘푗 is the torsion of a connection on 푀. 푇 = 푖 1 푖 푘 (푇푗푘) ∈ Ω푀(End 푇 푀). Locally 푇 = 푇푗푘푑푥 . Moerover given 푋, 푌 ∈ 푉 (푀), 휕 푇 (푋, 푌 ) = (푇 푖 푋푗푌 푘 ) ∈ 푉 (푀). 푗푘 휕푥푖 2 Then 푇 (푋, 푌 ) = −푇 (푌 , 푋) so 푇 ∈ Ω푀(푇 푀)

Definition (symmetric connection). A connection on a manifold 푀 is sym- metric if the torsion vanishes, i.e.

푖 푖 Γ푗푘 = Γ푘푗

in every local coordinates.

Denote the covariant derivative on 푀

0 1 퐷 ∶ Ω푀(푇 푀) = 푉 (푀) → Ω푀(푇 푀). 1 Denote by 퐷푋푌, where 푋, 푌 ∈ 푉 (푀) the evaluation of 퐷푌 ∈ Ω푀(푇 푀) on 푋. Thus 퐷푋푌 ∈ 푉 (푀). This is an R-linear map. In local coordinates,

푖 푗 푖 푖 푗 푘 (퐷푋푌 ) 휕푖 = (푋 휕푗푌 − Γ푗푘푌 푋 )휕푖 Consequently we obtain

Proposition 8.2. A connection 퐷 on 푀 is symmetric if and only if

퐷푋푌 − 퐷푌푋 = [푋, 푌 ]

for all 푋, 푌 ∈ 푉 (푀).

47 8 Riemannian geometry

Theorem 8.3 (Levi-Civita connection). On each Riemannian manifold (푀, 푔) there is a unique connection 퐷 such that

1. for every 푋, 푌 , 푍 ∈ 푉 (푀),

푍푔(푋, 푌 ) = 푔(퐷푍푋, 푌 ) + 푔(푋, 퐷푍푌 ).

2. 퐷 is symmetric. 퐷 is called Levi-Civita connection of 푔.

Recall that given 푋 ∈ 푉 (푀), 퐷푋 ∶ 푉 (푀) → 푉 (푀) is a covariant derivative if and only if

1. R-linearity in 푌: 퐷푋(휆푌 ) = 휆퐷푋푌 for all 휆 ∈ R, 푌 ∈ 푉 (푀), ∞ 2. Leibniz rule: 퐷푋(ℎ푌 ) = (푋ℎ) ⋅ 푌 + ℎ퐷푋푌 for all ℎ ∈ 퐶 (푀), ∞ ∞ 3. 퐶 (푀)-linearity in 푋: 퐷푓푋푌 = 푓퐷푋푌 for all 푓 ∈ 퐶 (푀). Proof of Theorem 8.3. We do uniqueness first. In a coordinate neighbourhood

푝 푘 퐷휕푖 = Γ푖푘휕푝푑푥 .

For condition 1 in the statement of the theorem, let 푋 = 휕푖, 푌 = 휕푗, 푍 = 휕푘 so that 푔(푋, 푌 ) = 푔푖푗, so we obtain the first equation

푝 푝 휕푘푔푖푗 = Γ푖푗푔푝푗 + Γ푗푘푔푖푝 푝 푝 휕푗푔푘푖 = Γ푘푗푔푝푖 + Γ푖푗푔푘푝 푝 푝 휕푖푔푗푘 = Γ푗푖푔푝푘 + Γ푘푖푔푗푝 the next two lines are similarly obtained by cyclic permutation on 푖, 푗, 푘. 푖푞 −1 Define (푔 ) = (푔푖푞) , the inverse matrix. Use the fact that 푔푖푞 is symmetric,

푝 푖푞 푖 Γ푗푘푔푝푞푔 = Γ푗푘.

Summing equations 1 + 2 − 3 gives

푝 휕푘푔푖푗 + 휕푗푔푘푖 − 휕푖푔푗푘 = 2Γ푗푘푔푖푝 which is the same as 1 Γ푝 푔 = (휕 푔 + 휕 푔 − 휕 푔 ). 푗푘 푞푝 2 푘 푞푗 푗 푘푞 푞 푗푘

Multiply by 푔푖푞 to get 1 Γ푖 = 푔푖푞(휕 푔 + 휕 푔 − 휕 푔 ) 푗푘 2 푘 푞푗 푗 푘푞 푞 푗푘 which is to say, there is at most one way to define the Christoffel symbols.

48 8 Riemannian geometry

Exercise. Show that the equation has a coordinate-free formulation 1 푔(퐷 푌 , 푍) = (푋 (푌 , 푍) + 푌 (푍, 푋) − 푍푔(푋, 푌 ) 푋 2 푔 푔 − 푔(푌 , [푋, 푍]) − 푔(푍, [푌 , 푋]) + 푔(푋, [푌 , 푍])).

To check the requirement for covariant derivative, 1 is clear from the coordinate- free formula. We omit the verification for 2 and 3 with the following checkpoints: 1. 3: Let 푓 ∈ 퐶∞(푀). Recall

[푓푋, 푍] = (푓푋)푍 − 푍(푓푋) = 푓[푋, 푍] − (푍푓)푋

so 1 푔(퐷 푌 , 푍) = (푓푋 (푌 , 푍) + 푌 (푓푔(푍, 푋)) − 푍(푓푔(푋, 푌 )) 푓푋 2 푔 − 푓(푔(푌 , [푋, 푍]) + 푔(푍, [푌 , 푋]) − 푔(푋, [푍, 푌 ])) + (푍푓)푔(푋, 푌 ) − (푌 푓)푔(푍, 푋) …

= 푔(푓퐷푋푌 , 푍)

then do some cancellation. We are also using

푌 (푓ℎ) = (푌 푓) ⋅ ℎ + 푓 ⋅ 푌 ℎ.

2. 2: let ℎ ∈ 퐶∞(푀), 1 푔(퐷 (ℎ푌 ), 푍) = (푋(ℎ푔(푌 , 푍)) + ℎ푌 (푔(푍, 푋)) − 푍(ℎ푔(푋, 푌 )) − ℎ푔(푌 , [푋, 푍]) − 푔(푍, [ℎ푌 , 푋]) + 푔(푋, [푍, ℎ푌 ]) 푋 2 1 = ((푋ℎ)푔(푌 , 푍) + ℎ푋푔(푌 , 푍) + ℎ푌 (푔(푍, 푋)) − (푍ℎ)푔(푋, 푌 ) − ℎ푍푔(푋, 푌 ) − ℎ푔(푌 , [푋, 푍]) + (푋ℎ)푔(푍, [푌 , 푋]) + (푍ℎ)푔(푋, 푌 ) + ℎ푔(푋, [푍, 푌 ]) 2

= (푋ℎ)푔(푌 , 푍) + ℎ푔(퐷푋푌 , 푍)

for all 푍. Thus 퐷푋(ℎ, 푌 ) = (푋ℎ)푌 + ℎ퐷푋푌 .

Thus 퐷 is a well-define connection. For statment 1 in theorem, we cantrace 푖 deduction of Γ푗푘 backwards.

8.1 Geodesics Let 훾 be a curve in 푈, a coordinate neighbourhood of 푀. Let 휋 ∶ 퐸 → 푀 be a 푖 vector bundle equipped with a connection 퐴, 퐴 = (Γ푗푘) on 푈. Then a lift of 훾 푘 to 퐸 is 훾퐸 such that 휋 ∘ 훾퐸 = 훾. If 훾(푡) = (푥 (푡)) in local coordinates, then

푘 푖 훾퐸(푡) = (푥 (푡), 푎 (푡)) in 푈 × 푉 = 휋−1(푈).A horizontal lift means that

̇훾 (푡) ∈ 푆 퐸 훾퐸(푡)

49 8 Riemannian geometry for all 푡 where 푆 denotes the horizontal subspace with respect to 퐴. In other words, 푖 휃 (퐸 ̇훾 (푡)) = 0 for 푖 = 1, … , 푚 where 푚 is the rank of 퐸. Expand out, 휕 휕 (푑푎푖 + Γ푖 푎푗푑푥푘)(푘 ̇푥 (푡) +푗 ̇푎 (푡) ) = 0. 푗푘 휕푥푘 휕푎푗 Simplify to get 푖 푖 푗 푘 ̇푎 + Γ푗푘(푥)푎 ̇푥 = 0. This is a system of 1st order linear ODEs so by basic results in ODE theorey, it is solvable on ant 퐼 ⊆ R and is uniquely determined by 푎푖(0). Thus horizontal lift always exists from each 푎 ∈ 퐸. Misses a lecture on 17/11/18 Fix 푝 ∈ 푀. If 훾(푡, 푎) = 훾푝(푡, 푎) is a geodesic, 휆 ∈ R then 푑 훾(휆푡, 푎) = 휆 ̇훾(휆푡, 푎) 푑푡 푑2 훾(휆푡, 푎) = 휆2 ̈훾(휆푡, 푎) 푑푡2 so 훾(휆푡, 푎) = 훾(푡, 휆푎). Furthermore

1. for all 푎 ∈ 푇푝푀, there exists 휀 = 휀푎 > 0 such that

훾(푠, 푎) = 훾(1, 푠푎)

exists for all |푠| < 휀.

2. from theory of ODEs, 휀푎 may be chosen continuous in 푎. For |푎|푔 ≤ 휀1, have 휀푎 ≥ 휀2 > 0. We have |푎|푔 < 휀 = 휀1휀2 then 훾(1, 푎) is defined.

Definition (exponential map). Let (푀, 푔) be a Riemannian manifold and 푝 ∈ 푀. The exponential map at 푝 is

exp푝 ∶ 푇푚푀 → 푀

푎 ↦ 훾푝(1, 푎)

Thus for some 휀 > 0 sufficiently small, exp푝 ∶ Ball푔(0, 휀) → 푀 is a well-defined smooth map.

Proposition 8.4. (푑 ) = exp푝 0 id푇푝푀

by noting 푇0(푇푝푀) ≅ 푇푝푀.

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Proof. One approach is to Taylor expand

2 훾(푡, 푎) = 푝 + 푎푡 + 푐2푡 + … and then substitute into geodesic ODE to get 1 푐푖 = Γ푖 (휙)푎푗푎푘. 2 2 푗푘 Then do some hands-on analysis.

Another approach is to note exp푝(0) = 푝. For |푎|푔 < 휀, 훾푝(푡, 푎) = 훾푝(1, 푡푎) for −1 ≤ 푡 ≤ 1. Then 푑 (푑 exp푝)0푎 = exp푝(푡푎)∣ = 0 푑푡 푡 푑 푑 = 훾푝(1, 푡푎)∣ = 훾푝(푡, 푎)∣ 푑푡 푡=0 푑푡 푡=0

=푝 ̇훾 (푡, 푎)∣ 푡=0 = 푎

Corollary 8.5. exp푝 ∶ Ball푔(0, 푟) → 푈 is a diffeomorphism onto its image 푈 for some 푟 > 0.

Proof. Inverse mapping theorem.

−1 This means that (exp푝) is a hcart around 푝 ∈ 푀. The respective local coordinates are the normal coordinates, aka geodesic coordinates at 푝. For exmaple in geodesic coordinates

−1 exp푝 (훾푝(푡, 푎)) = 푡푎 for |푡| sufficiently small. 푛 푛−1 The geodesic polars (푇푝푀, 푔(푝)) ≅ (R , Euclidean). Suppose (0, 휀) × 푆 is included in the domain of exp푝. Then can define a map

푓 ∶ (0, 휀) × 푆푛−1 → 푀

(푟, 푣) ↦ exp푝(푟푣)

푛−1 Let Σ푟 be the image under 푓 of {푟}×푆 , then 푓 defines an immersion (embed?) into 푀. This is a geodesic sphere around 푝.

Theorem 8.6 (Gauss lemma). 훾푝(푡, 푎) meets Σ푟 orthogonally for all 0 < 푟 < 휀. Thus in geodesic polars,

푔 = 푑푟3 + ℎ(푟, 푣)

푔 = 푔| where Σ푟 .

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So we may write 푔 as 푑푟2 0 … 0 ⎛ 0 ⎞ ⎜ ⎟ ⎜ ⋮ ℎ(푟, 푣) ⎟ ⎝ 0 ⎠

푛−1 푛−1 Proof. Choose any 푋 ∈ 푉 (푆 ). 푆 ⊆ 푇푝푀 with respect to 푔(푝). Extend 푋 ̃ o Ball1(0) \ {0} = {푎 ∈ 푇푝푀 ∶ |푎|푔 ≤ 1}. 푋(푟, 푣) = 푟푋(푣) on 퐵 \ {0}. Define ̃ 푌 (푓(푟, 푣)) = 푑(exp푝)푟푣푋(푟, 푣),

′ 휕 a vector field on 퐵 ⊆ exp푝(퐵 \ {0}) ⊆ 푀. To show 푌 ⟂ 휕푟 at each point in 퐵′ \ {푝}, note that 휕 1 =푝 ̇훾 (푡, 푎) 휕푟 |푎|푔(푝) ′ so ̇훾(푡, 푎) defines a vector field on 퐵 \ {푝} for all |푎|푔(푝) = 1 and 0 < 푡 < 휀. Take 휕 휕 limit as 푟 → 0 to obtain 푔( 휕푟 , 휕푟 ) = 1. Thus to show 푔(푌 , ̇훾) = 0, 휕 퐷 푌 − 퐷 ̇훾= (푑푓)(퐷 푋̃ − 퐷 ) ̇훾 푌 휕/휕푟 푋̃ 휕푟 푑 = (푑푓)( 푋)̃ 푑푟 푋̃ = (푑푓) 푟 푌 = 푟 by linearity of 푑푓 at each point. Thus 푑 푔(푌 , ̇훾) = 푔(퐷 푌 , ̇훾) + 푔(푌 , 퐷 )̇훾 푑푟 ̇훾 ⏟̇훾 =0 푌 = 푔(퐷 ̇훾+ , ̇훾) 푌 푟 1 = 푔(푌 , ̇훾) 푟 Let 퐺 = 푔(푌 , ̇훾) and we have an ODE 푑 퐺 퐺 = . 푑푟 푟 푑퐺 So 퐺 is linear in 푟 and 푑푟 is independent of 푟. But 푑퐺 휕 lim = lim 푔(푋, ) = 0 푟→표+ 푑푟 푟→0+ 휕푟 because (푑 exp)0 is an isometry.

8.2 Curvature of a Riemannian metrix

52 8 Riemannian geometry

Definition (Riemannian curvature). Let (푀, 푔) be a Riemannian manifold. The (full) Riemannian curvature 푅 = 푅(푔) of the metric 푔 is the curvature of the Levi-Civita connection of 푔.

2 By general theorey 푅(푔) ∈ Ω푀(End 푇 푀). In local coordinates, 1 푅 = 푅푖 푑푥푘 ∧ 푑푥ℓ 2 푗,푘ℓ

푘 where 푖, 푗, 푘, ℓ all take values 1, … , 푛 where 푛 = dim 푀. (푅푗,푘ℓ is called the Riemann curvature tensor. Let 푋, 푌 ∈ 푉 (푀). Then 푅(푋, 푌 ) ∈ Γ(End 푇 푀). Explicitly, if 푋 = 푘 ℓ 푋 휕푘, 푌 = 푌 휕ℓ then 푖 푘 ℓ 푅(푋, 푌 ) = (푅푗,푘ℓ푋 푌 )푖,푗.

We may define 푅푘ℓ = 푅(휕푘, 휕ℓ) to be local sections of End 푇 푀 so

푘 ℓ 푅(푋, 푌 ) = 푋 푌 푅푘ℓ.

푘 푖 푘 In local coordinates, 퐷 = 푑 + 퐴 where 퐴 = 퐴푘푑푥 = (Γ푗푘)푑푥 ). Denote 퐷 = 퐷 = 휕 + 퐴 . 푘 휕푘 푘 푘 For all 푍 ∈ 푉 (푀) have

푘 ℓ (−퐷 ∘ 퐷)푍 = 푅푍 = (푅푘ℓ푑푥 ∧ 푑푥 )푍

푅푘ℓ푍 = (퐷ℓ퐷푘 − 퐷푘퐷ℓ)푍 thus 푅푘ℓ = −[퐷푘, 퐷ℓ] and thus

푖 푖 푅푗,푘ℓ = ((퐷ℓ퐷푘 − 퐷푘퐷ℓ)휕푗)

푘 Write 퐷푋 = 푋 퐷푘 then we have

푘 ℓ −[퐷푋, 퐷푌] = −[푋 퐷푘, 푌 퐷ℓ] 푘 ℓ 푘 ℓ = −푋 (휕푘푌 )퐷ℓ − 푋 푌 퐷푘퐷ℓ 푘 ℓ 푘 ℓ + 푌 (휕푘푋 )퐷ℓ + 푌 푋 퐷ℓ퐷푘 푘 ℓ ℓ = 푋 푌 푅푘ℓ − [푋, 푌 ] 퐷ℓ

This proves

Lemma 8.7. 푅(푋, 푌 ) = 퐷[푋,푌 ] − [퐷푋, 퐷푌].

Sometimes it is convenient to consider

푞 푅푖푗,푘ℓ = 푔푖푞푅푗,푘ℓ by (푋, 푌 , 푍, 푇 ) ↦ 푔(푅(푋, 푌 )푍, 푇 ).

53 8 Riemannian geometry

Proposition 8.8 (symmetries of the curvature tensor).

1. 푅푖푗,ℓ푘 = −푅푖푗,푘ℓ = 푅푗푖,푘ℓ.

푖 푖 푖 2. 1st Bianchi identity: 푅푗,푘ℓ + 푅푘,ℓ푗 + 푅ℓ,푗푘 = 0.

3. 푅푖푗,푘ℓ = 푅푘ℓ,푖푗.

Proof. 1. The first equality is clear from properties of (“scalar”) 2-forms. Forthe second equality, 휕푔 푖푗 = 푔(퐷 휕 , 휕 ) + 푔(휕 , 퐷 휕 ) 휕푥푘 푘 푖 푗 푖 푘 푗 휕2푔 푖푗 = 푔(퐷 퐷 휕 , 휕 ) + 푔(퐷 휕 , 퐷 휕 ) + 푔(퐷 휕 , 퐷 휕 ) + 푔(휕 , 퐷 퐷 휕 ) 휕푥ℓ휕푥푘 ℓ 푘 푖 푗 푘 푖 ℓ 푗 ℓ 푖 푘 푗 푖 ℓ 푘 푗 so 휕2푔 휕2푔 0 = 푖푗 − 푖푗 휕푥ℓ휕푥푘 휕푥푘휕푥ℓ

= 푔([퐷ℓ, 퐷푘]휕푖, 휕푗) + 푔(휕푖, [퐷ℓ, 퐷푘]휕푗)

= 푔(푅푘ℓ휕푖, 휕푗) + 푔(휕푖, 푅푘ℓ휕푗)

= 푅푗푖,푘ℓ + 푅푖푗,ℓ

2.

푖 푖 푖 푖 푅푗,푘ℓ + 푅푘,ℓ푗 + 푅ℓ,푗푘 = [퐷ℓ퐷푘휕푗 − 퐷푘퐷ℓ휕푗 + 퐷푗퐷ℓ휕푘 − 퐷ℓ퐷푗휕푘 + two more terms]

But the first and last term cancel as [휕푘, 휕ℓ] = 0. 3. See the handout on octahedron trick.

푖 푗 Definition (Ricci curvature). The Ricci curvature of 푔 = 푔푖푗푑푥 푑푥 at 푝 ∈ 푀 is Ric푝(푋, 푌 ) = tr(푣 ↦ 푅푝(푋, 푣)푌 ).

Locally Ric = (Ric푖푗) so

푖 푗 Ric(푋, 푌 ) = Ric푖푗 푋 푌 .

We can also write 푞 푝푞 Ric푖푗 = 푅푖,푗푞 = 푔 푅푝푖,푗푞.

By the third identities we derive, Ric푖푗 is a symmetric bilinear form on 푇푝푀 for all 푝.

54 8 Riemannian geometry

Definition (scalar curvature). The scalar curvature 푠 = scal(푔) ∈ 퐶∞(푀) is the trace of Ric with respect to 푔, i.e. if 푔푖푗(푝) = 훿푖푗 then 푠 = Ric푖푖.

In general, 푖푗 푖푗 푝푞 푗푘 ℓ 푠 = 푔 Ric푖푗 = 푔 푔 푅푝푖,푗푞 = 푔 푅푗,푘ℓ. We can show, using linear algebra and representation theory, that Ric and 푠 are the only 푆푂(푛)-invariant components in the “space of curvature tensor”. Missed a lecture on 24/11/18 (Laplacian, Hodge star operator) Recall that 훿 = (−1)푛(푝+1)+1 ∗푑∗ on 푝-forms for 푝 ≥ 1 and 훿 = 0 on functions. Proof. Given 훼 ∈ Ω푝−1(푀), 훽 ∈ Ω푝(푀), 푑(훼 ∧ ∗훽) = 푑훼 ∧ ∗훽 + (−1)푝−1훼 ∧ 푑(∗훽)

Now

− ∗ 훿훽 = (−1)푛(푝+1) ∗ ∗ 푑⏟ ∗ 훽 ∈Ω푛−푝+1 = (−1)푛(푝+1)+(푛−푝+1)(푝−1)푑 ∗ 훽 = (−1)−1푑 ∗ 훽 (mod 2) Thus 푑(훼 ∧ ∗훽) = 푑훼 ∧ ∗훽 − 훼 ∧ ∗훿훽 = ⟨푑훼, 훽⟩푔휔푔 − ⟨훼, 훿훽⟩푔휔푔 Now proposition follows from Stokes’. Remark. We can define for 휁, 휂 ∈ Ω푝(푀)

⟨⟨휁, 휂⟩⟩푀,푔 = ∫ ⟨휁, 휂⟩푔휔푔 푀 the 퐿2 inner product of 푝-forms. The the proposition asserts that ⟨⟨푑훼, 훽⟩⟩ = ⟨⟨훼, 훿훽⟩⟩ i.e. 훿 = 푑∗, the formal 퐿2-adjoint of 푑.

Corollary 8.9. Δ = 푑훿 + 훿푑 is formally self-adjoint.

Definition (Laplace-Beltrami/Hodge Laplacian). Δ is called Laplace-Beltrami, also known as the Hodge Laplacian.

We give a special name to kernel of Δ:

Definition (harmonic 푝-form). The space of harmonic 푝-forms is defined as ℋ푝(푀) = {훼 ∈ Ω푝(푀) ∶ Δ훼 = 0}.

Note. ∗Δ = Δ∗ so ∗ ∶ ℋ푝(푀) → ℋ푛−푝(푀) is an isomorphism.

55 8 Riemannian geometry

Proposition 8.10. Assume 푀 is compact. Then for all 훼 ∈ Ω푝(푀), Δ훼 = 0 if and only if 푑훼 = 0 and 훿훼 = 0.

The condition 훿훼 = 0 is called 훼 is coclosed. Proof. • ⟹ : 0 = ⟨⟨Δ훼, 훼⟩⟩ = ⟨⟨푑훿훼, 훼⟩⟩ + ⟨⟨훿푑훼, 훼⟩⟩ 2 2 = ‖훿훼‖퐿2 + ‖푑훼‖퐿2 but for all 휁 ∈ Ω∗(푀),

‖휁‖퐿2 = √⟨⟨휁, 휁⟩⟩ ≥ 0 with equality if and only if 휁 = 0 as 휁 is smooth. Thus 훿훼 = 0, 훿훼 = 0. • ⟸ : obvious.

Corollary 8.11. Let 푀 be compact and connected. Then if 푓 ∈ 퐶2(푀) is harmonic it is constant. Note. Compactness is essential. For example (푥, 푦) ↦ 푒푦 cos 푥 on the Euclidean space R2. Note that on (R푛, 푔) where 푔 is not necessarily Euclidean, 1 휕 휕푓 Δ = −√ (√det 푔푔푖푗 ). 푔 det 푔 휕푥푗 휕푥푖 When 푔 is Euclidean, this simplies to 휕2 Δ = − ∑ . Eucl (휕푥푖)2 It is an exercise to check that this has positive eigenvalues.

Theorem 8.12 (Hodge decomposition theorem). Let (푀, 푔) be a compact oriented Riemannian manifold. Let 0 ≤ 푝 ≤ dim 푀. Then

1. dim ℋ푝(푀) is finite. 2.

Ω푝(푀) = ℋ푝(푀) ⊕ ΔΩ푝(푀) = ℋ푝(푀) ⊕ 푑Ω푝(푀) ⊕ 훿푑Ω푝(푀) = ℋ푝(푀) ⊕ 푑Ω푝−1(푀) ⊕ 훿Ω푝+1(푀)

where all the direct sums are orthogonal with respect to 퐿2.

56 8 Riemannian geometry

Proof. Too long. Omitted.

Corollary 8.13. Assume the hypothesis of Hodge decomposition theorem, 푝 푝 then for all 푎 ∈ 퐻dR(푀) there exists a unique 훼 ∈ ℋ (푀) such that [훼] = 푎.

푝 Proof. We do uniqueness first: suppose 훼1, 훼2 ∈ ℋ (푀) with 훼1 − 훼2 = 푑훽. Then 2 ‖푑훽‖ = ⟨⟨푑훽, 훼1 − 훼2⟩⟩ = ⟨⟨훽, 훿훼⏟⏟⏟⏟⏟1 − 훿훼2⟩⟩ = 0 0 so 푑훽 = 0 and 훼1 = 훼2. For existence, suppose 푎 = [ ̃훼] where 푑 ̃훼= 0 but ̃훼 is not necessarily closed. By Hodge decomposition theorem we can write ̃훼= 훼+푑훽 +훿훾.

Then 푑훿훾 = 0 so 0 = ⟨⟨푑훿훾, 훾⟩⟩ = ‖훿훾‖2 so 훿훾 = 0. It follows that 훼 ∈ ℋ푝(푀). What we have shown is that

푝 푝 ℋ (푀) → 퐻dR(푀) 훼 ↦ [훼] is a linear isomorphism. LHS is an analytic object, and by de Rham theory, RHS is the same as 퐻푝(푀), a topological object. The significance of Hodge decomposition is this correspondence. 푝 An intuitive explanation of the corollary: given 푎 ∈ 퐻dR(푀), define 푝 퐵푎 = {휁 ∈ Ω (푀) ∶ 푑휁 = 0, [휁] = 푎}

2 and consider the function 퐹 (휁) = ‖휁‖ on 퐵푎. If 훼 is a minimum of 퐹 the 푑 ∣ 퐹 (훼 + 푡푑훽) = 0 푑푡 푡=0 for all 훽 so 푑 ∣ = ⟨⟨훼 + 푡푑훽, 훼 + 푡푑훽⟩⟩ 푑푡 푡=0 = 2⟨⟨훼, 푑훽⟩⟩ 훿훼 = 0 = 2⟨⟨훿훼, 훽⟩⟩ = 0 so 훼 ∈ ℋ푝(푀). A remark about the second statement of Hodge decomposition theorem: the last two lines follow from the first for similar reason to the corollary. If Δ훼 = 훽 then 훽 ⟂ ℋ푝(푀) so (ℋ푝(푀))⟂ is well-defined and the projection of 훽 to this space is given by dim ℋ푝(푀)

훽 − ∑ ⟨⟨훽, 휔푖⟩⟩휔푖 푖=1

57 8 Riemannian geometry

푝 푝 푝 ⟂ where {휔푖} is an orthonormal basis of ℋ (푀). Thus ΔΩ (푀) ⊆ (ℋ (푀)) . We can solve for 휔 is Δ휔 = 훼, 훼 ⟂ ℋ푝. First define a “weak solution” of Δ휔 = 훼, a bounded linear functional on Ω푝(푀). Then argue using a regularity theorem from analysis every weak solution is an actual solution. The key idea is that Δ is elliptic.

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