Affine Connections: Part 2

Home , Connection form, Ehresmann connection, Parallel transport

Aﬃne Connections: Part 2 Manuscript for Machine Learning Reading Group Talk ∗

R. Simon Fong †

Abstract

Note for online manuscript: This is the manuscript of a one hour introductory talk on (aﬃne) connections. This is intended to be a supplement to the talk, hence discussions are fairly brief and far from complete. Since some (a lot of!) deﬁnitions are omitted for brevity, readers are encouraged to refer to classical texts such as J. M. Lee’s smooth manifold trilogy for a more formal and complete discussions [4, 5].

The abstract of the talk is as follows: We continue our discussions on (aﬃne) connections on Riemannian manifolds.

Last time we discussed the necessary preliminaries (and hopefully motivations) to study/establish connections on ﬁbre bundles. The goal of this talk is to establish parallel transport as a natural way to relate local geometrical structures via connection.

In the second half we will discuss a more general approach from the other end of the spectrum via connections on ﬁbre bundles - Ehresmann connections.

Please note that some diagrams presented in the talk will not be included due to the author’s artistic skills (or the lack thereof).

Some formulas and notations are expanded slightly from the previous talk.

Since we didn’t discuss Einstein summation convention last time, we won’t use it here. In most cases it is the more convenient notation.

∗The talk was given in the Machine Learning reading group of School of Computer Science at University of Birmingham on 16th March 2016. †School of Computer Science, University of Birmingham, Birmingham B15 2TT, United Kingdom

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Contents

5 A quick recap 3 5.1 Assumptions ...... 3 5.2 Recall: from Part 1 ...... 3

6 Connections and parallel transport 4 6.1 Connection on Vector Bundles ...... 4 6.2 Parallel Transport ...... 6

7 Fibre bundle connection 8 7.1 Recall: Fibre Bundles ...... 8 7.2 Bundle map ...... 9 7.3 Ehresmann connection ...... 10 7.4 Horizontal lift ...... 11 7.4.1 Back to the notion of “parallel” ...... 11

References 12

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5 A quick recap

5.1 Assumptions • All manifolds are abstract topological spaces. We do not assume them to be subsets of some ambient Euclidean space. • All manifolds and functions are assumed to be smooth.

5.2 Recall: from Part 1 Previously we discussed: that geometrical properties of manifolds are inherited only locally from Euclidean spaces. Even though each ﬁbre of the tangent bundle is homeomorphic (linearly isomorphic) to Rk), there is no natural homeomorphism between ﬁbres of tangent bundle. If we restrict our attention to M = Rn, we observed: n n n ∼ n 1. Geometric tangent spaces of R behaves nicely: for any p ∈ R , TpR = R .

n n 2. R admits a global frame: namely the standard basis {ei} of T0R .) These two observations allows us to make the following deﬁnition in vector calculus (using the terminologies of manifolds):

Deﬁnition 5.1. The covariant directional derivative of vector ﬁelds in Rn is the bilinear map:

n n n ∇ : T R × T R → T R (X,Y ) 7→ ∇X Y The map ∇ satisﬁes the following properties:

1. Covariant (a.k.a. tensorial with respect to direction OR linear in C∞(Rn) in X in some literatures OR C∞(Rn)-linear)

∞ n n ∇f·X1+g·X2 Y = f · ∇X1 Y + g · ∇X2 Y, ∀f, g ∈ C (R ), ∀X1,X2 ∈ T R

2. Linear over R in Y

n n ∇X aY1 + bY2 = a · ∇X Y1 + b · ∇X Y2, ∀a, b ∈ R , ∀Y1,Y2 ∈ T R

3. Leibniz (product) rule:

∞ n ∇X (f · Y ) = DX f · Y + f · ∇X Y, ∀f ∈ C (R )

Remark 5.2. Given a vector ﬁeld Y ∈ T Rn, property 1 implies it only depends on the direction (given by X) at a point. ♦

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Given a vector ﬁeld V on Rn, there is a natural way to construct such a covariant direction derivative by:

Vp+t·Xp − Vp ∇X V | = lim (1) p t→0 t In abstract manifold we run into two problems (number 1, 2 corresponds to our earlier observations 1, 2 of Rn respectively):

1. What does it mean by Vp+t·Xp , speciﬁcally, what does the subscript p + t · Xp mean? 2. Tangent spaces are disjoint, in other words we can’t do the following quotient:

Vp+t·Xp − Vp

Nevertheless this gives us an idea to “relate local geometry information" by finding a specific way to map one tangent space to another via a covariant derivative of vectors fields (fibres). Hence the notion and the name of connections.

6 Connections and parallel transport

6.1 Connection on Vector Bundles We wish to mimic the previous discussion and construct something similar on vector bundles over abstract (smooth) manifolds, hence we construct the following map [5]: Deﬁnition 6.1. Let π : E → M be a vector bundle over M, a connection in E is the map:

∇ : T M × E(M) → E(M) where E(M) is the smooth sections of E, such that ∇ satisﬁes: 1. C∞(M)-linear in X

∞ ∇f·X1+g·X2 Y = f · ∇X1 Y + g · ∇X2 Y, ∀f, g ∈ C (M), ∀X1,X2 ∈ T M

2. Linear over R in Y

n ∇X aY1 + bY2 = a · ∇X Y1 + b · ∇X Y2, ∀a, b ∈ R , ∀Y1,Y2 ∈ E(M)

3. Leibniz (product) rule:

∞ ∇X (f · Y ) = Xf · Y + f · ∇X Y, ∀f ∈ C (M)

Remark 6.2. 1. We call ∇X Y the covariant derivative of Y in direction X.

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2. For p ∈ M, ∇X Y depends (only) on Y on some neighbourhood of p, and X at p. ♦

Remark 6.3. In algebraic geometry [1], connections are sometimes deﬁned equivalently as the map:

∇ : E(M) → E(Λ1(M) ⊗ M) where Λ1(M) denote 1−forms of M. Notice this conversion is made by taking the direction input (from T (M)) to the output, and observe that Im(∇) are (smooth) sections of a tensor bundle.

• For the rest of our discussion we will stick to the map in Deﬁnition 6.1.

♦

Restricting our discussion to the tangent bundle TM over M, we obtain:

Deﬁnition 6.4. An aﬃne connection on M is the connection in TM:

∇ : T M × T M → T M where T M are smooth sections of tangent bundles, i.e. smooth vector ﬁelds on M.

Remark 6.5. Let U be an open subset of M, suppose {Ei} is a local frame (linearly independent sections) of TM on U. For each pair of indices i, j, we can express ∇Ei Ej by:

X k ∇Ei Ej = Γi,jEk i,j

k 3 Γi,j is a set of n functions called the Christoffel symbols (of the second kind). Turns out Affine connections are completely described by Christoffel symbols: Given U ⊂ M, again {Ei} be a local frame (linearly independent sections) of TU such that P i P j X,Y ∈ T U can be expressed as i X Ei, j Y Ej respectively, then

X i k i j k ∇X Y = X EiY + X Y Γi,j Ek (2) i,j,k

♦

In particular when we look at M = Rn, the Euclidean connection is given by:

X j ∇X Y = XY Ej j

In other words, the Christoﬀel symbols vanish identically in standard coordinates. To express ∇X Y in the form of 1, we need to speciﬁc one way to relate the local tangent spaces.

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6.2 Parallel Transport

Definition 6.6. A vector field V ∈ T M is parallel if ∇X V ≡ 0 for all X ∈ T M Whilst nonzero parallel vector fields don’t exist in general, parallel vector fields along a curve do. Definition 6.7. Given a curve γ : I → M, a vector field along γ is a (smooth) map V : I → TM such that V (t) ∈ Tγ(t)M. We further assume all curves γ to be injective.

Deﬁnition 6.8. A vector ﬁeld V along γ : I → M is parallel along γ if ∇γ˙ (t)V ≡ 0 for P d i all t ∈ I. γ˙ (t) := i dt γ (t)Ei for some local frame {Ei}

Figure 1: [Parallel Vector ﬁeld along curve in R2]

Theorem 6.9. Given a curve γ : I → M, t0 ∈ I, a vector V0 ∈ Tγ(t0)M, there exists a unique parallel vector ﬁeld V along γ, such that V (t0) = V0. Remark 6.10. 1. The proof is given by Picard-Lindelöf theorem (existence and uniqueness of linear ODE solutions). Uniqueness comes from the fact that we require the extension to be parallel along γ.

2. V is called parallel translation of V0 along γ. ♦ Parallel translation deﬁnes an important operator: the natural linear isomorphism between tangent spaces.

Deﬁnition 6.11. Given a curve γ : I → M, t0, t1 ∈ I, parallel transport from Tγ(t0 M to Tγ(t1 M is the linear isomorphism:

Pt0,t1 : Tγ(t0)M → Tγ(t1)M

such that given a vector V0 ∈ Tγ(t0)M, for any t1 ∈ I:

Pt0,t1 V0 = Pt0,t1 V (t0) = V (t1) where V is the parallel translation of V0 along γ. Finally, we retrieve a formula of covariant derivatives in M very much similar to that one we deﬁned in Rn:

Lemma 6.12. Let V ∈ T (γ) be a vector ﬁeld along γ. The covariant derivative ∇γ˙ (t)V (t) along γ can be expressed as:

P −1V (t) − V (t ) t0,t 0 ∇γ˙ (t)V (t) = lim (3) t=t0 t→t0 t − t0

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Proof. Let V ∈ T (γ) be a vector ﬁeld along γ. Suppose in some neighbourhood of γ(t0), local coordinates are denoted by {xi}, then we can write:

X j V (t) = V (t)∂j j

n ∂ o where {∂j} = is a local frame near p = γ(t0). ∂xj n ˜ o By theorem 6.9, we extend {∂j} to a a parallel frame (of vector ﬁelds) ∂j(t) along γ. ˜ ˜ Moreover ∂j(t) are parallel implies ∇γ(t)∂j ≡ 0. Hence we have the following expansion (by equation 2):

X ˙ j j ∇γ˙ (t)V (t) = V (t0)∂j + V (t0) ∇γ(t˙ )∂j t=t0 0 j | {z } = 0 X ˙ j = V (t0)∂j j j j j j X V (t) − V (t0) X V (t)∂j − V (t0)∂j = lim ∂j = lim t→t t→t j 0 t − t0 j 0 t − t0 P −1V j(t)∂˜ (t) − V j(t )∂ = X lim t0,t j 0 j t→t j 0 t − t0 P −1V (t) − V (t ) = lim t0,t 0 t→t0 t − t0 where

j 1. ﬁrst row is by Leibniz rule (V (t0) are smooth real valued functions) ˜ ˜ 2. second row is by the fact that ∂j’s are parallel, and ∂j = ∂j(t0) (as indicated by the underbrace)

3. third row is just deﬁnition of derivative of real valued functions

4. forth row is by deﬁnition of parallel transport:

˜ ˜ −1 ˜ ∂j(t) = Pt0,t∂j(t0) = Pt0,t∂j ⇐⇒ Pt0,t∂j(t) = ∂j

Remark 6.13. Notice equation 3 in lemma 6.12 is very similar to equation 1, which is exactly what we wanted. ♦

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Figure 1: Parallel translation of vector ﬁeld A along curves on a sphere 1

So far we restricted our discussion to vector (tangent) bundles, we now approach it from the other end of the spectrum.

7 Fibre bundle connection

7.1 Recall: Fibre Bundles Definition 7.1. Given a topological space M, a fibre bundle over M is the structure (E, M, π, F ). E is a topological space called the total space, M the base, F another topological space called the fibre, and a continuous surjection π : E → M called the projection. The structure (E, M, π, F ) satisfies

−1 1. For each p ∈ M, Ep := π (p) is homeomorphic to F . 2. Moreover, ∀p ∈ M, ∃U neighbourhood of p, such that the following diagram commutes with homeomorphism (local trivialization) ϕ : π−1(U) → U × Rk

ϕ π−1(U) ⊂ E U × F

π π1 U ⊂ M

−1 k In particular ϕp : π (p) = Ep → {p} × R is a homeomorphism. A ﬁbre bundle is smooth if all the spaces and maps are smooth. In particular the local trivialization is a diﬀeomorphism.

1Image from wikipedia

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Example 7.2. Trivial Bundle If E = M × F and π : E → M is the natural projection onto M, then (E, M, π, F ) is called the trivial bundle. In fact the term local trivialization just means the bundle “looks like” the trivial bundle locally. N Deﬁnition 7.3. A (smooth) section of E is a (smooth) continuous map σ : M → E such that π ◦ σ = IdM . Equivalently σ(p) ∈ Ep for all p. Remark 7.4. If the context is clear, we often refer to ﬁbre bundle (E, M, π, F ) as one of the following: 1. π : E → M 2. π 3. E

And we often refer to Ep = Eπ(u) as ﬁbres as well. ♦

7.2 Bundle map

Deﬁnition 7.5. Given two ﬁbre bundles πM : EM → M, πN : EN → N and a continuous map F : M → N.A bundle map from M to N is a pair of continuous maps (F,F∗) such that the diagram commutes:

F∗ EM EN

πM πN M F N

Hence F ◦ πM = πN ◦ F∗, and F∗ is fibre preserving. We may refer the bundle map by F∗ and say F∗ covers F . F∗ is often refered to as the tangent map, differential, or pushforward (in the context of tangent bundles) of F . For the rest of the talk we will call it the tangent map of F to avoid confusion with objects like differential forms. In cases where EM ,EN are vector bundles, this is often called a vector bundle homomorphism.

Remark 7.6. In the context of tangent bundles πM : TM → M and πN : TN → N. Given continuous map F : M → N, then we can deﬁne the pushforward F∗ : TM → TN associated with F . For each p ∈ M, the pushforward of F is given by:

(F∗X)(f) = X(f ◦ F )

∞ where X ∈ TpM, f ∈ C (N), and F∗X ∈ TF (p)N. Note that pushforward of a tangent vector doesn’t always exist. If F is a smooth map between smooth manifolds, then F∗ is also smooth, and so (F,F∗) is a smooth bundle map. ♦

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7.3 Ehresmann connection One might notice that we run into the same problem once again in fibre bundles: although each fibre Ep = Eπ(u) is homeomorphic to F , there is no natural homeomorphism between fibres. Therefore a general notion of connection is necessary [2]:

Definition 7.7. Let (E, M, π, F ) be a smooth fibre bundle2, the vertical bundle V is the subbundle defined by:

V := ker(π∗) = ker(π∗ : TE → TM)

For each u ∈ E, we have Vu := ker(π∗ : TuE → Tπ(u)M) = Tu(Eπ(u)). An Ehresmann connection on the ﬁbre bundle E is a smooth subbundle H com- plementary to V in the sense that TE = V ⊕ H [3]. In other words it is a collection of subspaces H := {Hu ⊂ TuE | u ∈ E} such that TuE = Vu ⊕ Hu for all u ∈ E. H is also called the horizontal subbundle, and Hu the horizontal subspaces. Each horizontal subspace Hu ∈ H also satisﬁes:

1. For each u ∈ E, Hu is a vector subspace of the tangent space TuE.

2. u 7→ Hu is smooth We can view it as the bundle map:

TE π∗

E TM π

M where each “horizontal” and “vertical” components are ﬁbre bundles.

Figure 2: [Ehresmann connection on line bundle]

Remark 7.8. π∗ : TuE → Tπ(u)M, hence Im(π∗) = TM. So one can think of vectors in the kernel (elements of Vu) as “directions” (in the same sense that derivations or tangent vectors are “directions”) within fibres in the fibre bundle. Hence (somewhat loosely) vectors in the image (elements of Hu) can be somewhat considered as “directions complement to staying inside” ↔ “directions not staying inside” ↔“directions through fibres”. ♦

2This also works for ﬁbred manifolds, i.e. when we don’t have a typical ﬁbre F , and π is a surjective submersion .

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7.4 Horizontal lift To retrieve a similar notion of parallel transport transport we require an extra deﬁnition.

Deﬁnition 7.9. Given a curve γ : I → M.A lift of γ to E is the curve γ˜ : I → E such that for all t ∈ I:

π(˜γ(t)) = γ(t)

In other words, the following diagram commutes:

E γ˜ π γ I M

A lift is horizontal if for all t, γ˜˙ (t) belongs to the horizontal subspace:

γ˜˙ (t) ∈ Hγ˜(t) ⊂ Tγ˜(t)E

Remark 7.10. Suppose γ(t0) = p, then each u ∈ Eγ(t0) = Ep = Eπ(u) defines a choice of lift with γ˜(t0) = u , which we call lift of γ through u. ♦ We thus obtain a similar notion of parallel transport in fibre bundles (for a sufficiently small time t) [7]:

Theorem 7.11. Given a ﬁbre bundle (E, M, π, F ), and Ehresmann connection H. Let p ∈ M, and γ : I → M be a curve through p such that γ(t0) = p. For each u ∈ Ep, there is a unique horizontal lift of γ through u for amount of small time t.

7.4.1 Back to the notion of “parallel”

Using the direct sum decomposition of TE: TuE = Vu ⊕ Hu for all u ∈ E, we can alternatively deﬁne Ehresmann connection using connection form v, where v is the projection onto the vertical subbundle given by the vector bundle homomorphism:

v : TE → TE

TuE 7→ Vu

The horizontal subbundle can therefore be expressed alternatively as

H = ker v n o = H ⊂ T E | H = ker v| u u u TuE

Hence if γ˜(t) is a horizontal curve, X := γ˜˙ (t) ∈ H implies v| X ≡ 0. This is γ˜(t) Tγ˜(t)E similar to the notion of parallel vector ﬁelds where ∇V ≡ 0.

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References

[1] P. Griﬃths and J. Harris. Principles of algebraic geometry. John Wiley & Sons, 2014.

[2] S. Kobayashi and K. Nomizu. Foundations of diﬀerential geometry, volume 1. New York, 1963.

[3] I. Kolár, J. Slovák, and P. W. Michor. Natural operations in diﬀerential geometry. 1999.

[4] J. M. Lee. Smooth manifolds. Springer, 2003.

[5] J. M. Lee. Riemannian manifolds: an introduction to curvature, volume 176. Springer Science & Business Media, 2006.

[6] J. M. Lee. Introduction to topological manifolds, volume 940. Springer Science & Business Media, 2010.

[7] M. Spivak. A comprehensive introduction to diﬀerential geometry. Vol. II. Publish or Perish Inc., Wilmington, Del., second edition, 1979.