Affine Connections: Part 2

Transcription

1 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 16 th March School of Computer Science, University of Birmingham, Birmingham B15 2TT, United Kingdom 1

2 Contents 5 A quick recap Assumptions Recall: from Part Connections and parallel transport Connection on Vector Bundles Parallel Transport Fibre bundle connection Recall: Fibre Bundles Bundle map Ehresmann connection Horizontal lift Back to the notion of parallel References 12 2

3 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 R k ), there is no natural homeomorphism between fibres of tangent bundle. If we restrict our attention to M = R n, we observed: 1. Geometric tangent spaces of R n behaves nicely: for any p R n, T p R n = R n. 2. R n admits a global frame: namely the standard basis {e i } of T 0 R n.) 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 R n bilinear map: is the : T R n T R n T R n The map satisfies the following properties: (X, Y ) X Y 1. Covariant (a.k.a. tensorial with respect to direction OR linear in C (R n ) in X in some literatures OR C (R n )-linear) f X1 +g X 2 Y = f X1 Y + g X2 Y, f, g C (R n ), X 1, X 2 T R n 2. Linear over R in Y X ay 1 + by 2 = a X Y 1 + b X Y 2, a, b R n, Y 1, Y 2 T R n 3. Leibniz (product) rule: X (f Y ) = D X f Y + f X Y, f C (R n ) Remark 5.2. Given a vector field Y T R n, property 1 implies it only depends on the direction (given by X) at a point. 3

4 Given a vector field V on R n, there is a natural way to construct such a covariant direction derivative by: X V p = lim t 0 V p+t Xp V p t In abstract manifold we run into two problems (number 1, 2 corresponds to our earlier observations 1, 2 of R n respectively): 1. What does it mean by V p+t Xp, specifically, what does the subscript p + t X p mean? 2. Tangent spaces are disjoint, in other words we can t do the following quotient: V p+t Xp V p 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 X 2 Y = f X1 Y + g X2 Y, f, g C (M), X 1, X 2 T M 2. Linear over R in Y X ay 1 + by 2 = a X Y 1 + b X Y 2, a, b R n, Y 1, Y 2 E(M) 3. Leibniz (product) rule: X (f Y ) = Xf Y + f X Y, f C (M) Remark We call X Y the covariant derivative of Y in direction X. 4 (1)

5 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(Λ 1 (M) M) where Λ 1 (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 T M over M, we obtain: Definition 6.4. An affine connection on M is the connection in T M: : 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 {E i } is a local frame (linearly independent sections) of T M on U. For each pair of indices i, j, we can express Ei E j by: Ei E j = i,j Γ k i,je k Γ k i,j is a set of n 3 functions called the Christoffel symbols (of the second kind). Turns out Affine connections are completely described by Christoffel symbols: Given U M, again {E i } be a local frame (linearly independent sections) of T U such that X, Y T U can be expressed as i X i E i, j Y j E j respectively, then X Y = i,j,k ( X i E i Y k + X i Y j Γ k i,j) Ek (2) In particular when we look at M = R n, the Euclidean connection is given by: X Y = j XY j E 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. 5

6 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 T M such that V (t) T γ(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 all t I. γ(t) := i d dt γi (t)e i for some local frame {E i } Figure 1: [Parallel Vector field along curve in R 2 ] Theorem 6.9. Given a curve γ : I M, t 0 I, a vector V 0 T γ(t0 )M, there exists a unique parallel vector field V along γ, such that V (t 0 ) = V 0. Remark 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 V 0 along γ. Parallel translation defines an important operator: between tangent spaces. the natural linear isomorphism Definition Given a curve γ : I M, t 0, t 1 I, parallel transport from T γ(t0 M to T γ(t1 M is the linear isomorphism: P t0,t 1 : T γ(t0 )M T γ(t1 )M such that given a vector V 0 T γ(t0 )M, for any t 1 I: P t0,t 1 V 0 = P t0,t 1 V (t 0 ) = V (t 1 ) where V is the parallel translation of V 0 along γ. Finally, we retrieve a formula of covariant derivatives in M very much similar to that one we defined in R n : Lemma Let V T (γ) be a vector field along γ. The covariant derivative γ(t) V (t) along γ can be expressed as: γ(t) V (t) Pt 1 t=t0 = lim 0,tV (t) V (t 0 ) (3) t t0 t t 0 6

7 Proof. Let V T (γ) be a vector field along γ. Suppose in some neighbourhood of γ(t 0 ), local coordinates are denoted by {x i }, then we can write: V (t) = j V j (t) j where { j } = { x j } is a local frame near p = γ(t0 ). 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): γ(t) V (t) t=t0 = j = j V j (t 0 ) j + V j (t 0 ) γ(t 0 ) j }{{} = 0 V j (t 0 ) j = j = j V j (t) V j (t 0 ) lim j = t t 0 t t 0 j Pt 1 lim 0,tV j (t) j (t) V j (t 0 ) j t t 0 t t 0 Pt 1 = lim 0,tV (t) V (t 0 ) t t0 t t 0 V j (t) j V j (t 0 ) j lim t t 0 t t 0 where 1. first row is by Leibniz rule (V j (t 0 ) are smooth real valued functions) 2. second row is by the fact that j s are parallel, and j = j (t 0 ) (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: j (t) = P t0,t j (t 0 ) = P t0,t j P 1 t 0,t j (t) = j Remark Notice equation 3 in lemma 6.12 is very similar to equation 1, which is exactly what we wanted. 7

8 N S 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. For each p M, E p := π 1 (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 R k π 1 (U) E ϕ U F π π 1 U M In particular ϕ p : π 1 (p) = E p {p} R k is a homeomorphism. A fibre bundle is smooth if all the spaces and maps are smooth. In particular the local trivialization is a diffeomorphism. 1 Image from wikipedia 8

9 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. Definition 7.3. A (smooth) section of E is a (smooth) continuous map σ : M E such that π σ = Id M. Equivalently σ(p) E p 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 E p = E π(u) as fibres as well. 7.2 Bundle map Definition 7.5. Given two fibre bundles π M : E M M, π N : E N 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: E M F E N π M π N M F Hence F π M = π N 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 E M, E N are vector bundles, this is often called a vector bundle homomorphism. Remark 7.6. In the context of tangent bundles π M : T M M and π N : T N N. Given continuous map F : M N, then we can define the pushforward F : T M T N associated with F. For each p M, the pushforward of F is given by: N (F X) (f) = X(f F ) where X T p M, f C (N), and F X T F (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. 9

10 7.3 Ehresmann connection One might notice that we run into the same problem once again in fibre bundles: although each fibre E p = E π(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 bundle 2, the vertical bundle V is the subbundle defined by: V := ker(π ) = ker(π : T E T M) For each u E, we have V u := ker(π : T u E T π(u) M) = T u (E π(u) ). An Ehresmann connection on the fibre bundle E is a smooth subbundle H complementary to V in the sense that T E = V H [3]. In other words it is a collection of subspaces H := {H u T u E u E} such that T u E = V u H u for all u E. H is also called the horizontal subbundle, and H u the horizontal subspaces. Each horizontal subspace H u H also satisfies: 1. For each u E, H u is a vector subspace of the tangent space T u E. 2. u H u is smooth We can view it as the bundle map: T E E π π T M M where each horizontal and vertical components are fibre bundles. Figure 2: [Ehresmann connection on line bundle] Remark 7.8. π : T u E T π(u) M, hence Im(π ) = T M. So one can think of vectors in the kernel (elements of V u ) 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 H u ) can be somewhat considered as directions complement to staying inside directions not staying inside directions through fibres. 2 This also works for fibred manifolds, i.e. when we don t have a typical fibre F, and π is a surjective submersion. 10

11 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: I γ γ E M π A lift is horizontal if for all t, γ(t) belongs to the horizontal subspace: γ(t) H γ(t) T γ(t) E Remark Suppose γ(t 0 ) = p, then each u E γ(t0 ) = E p = E π(u) defines a choice of lift with γ(t 0 ) = 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 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(t 0 ) = p. For each u E p, there is a unique horizontal lift of γ through u for amount of small time t Back to the notion of parallel Using the direct sum decomposition of T E: T u E = V u H u 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 : T E T E T u E V u The horizontal subbundle can therefore be expressed alternatively as H = ker v = { H u T u E H u = ker v TuE Hence if γ(t) is a horizontal curve, X := γ(t) H γ(t) implies v T γ(t) E X 0. This is similar to the notion of parallel vector fields where V }

12 References [1] P. Griffiths and J. Harris. Principles of algebraic geometry. John Wiley & Sons, [2] S. Kobayashi and K. Nomizu. Foundations of differential geometry, volume 1. New York, [3] I. Kolár, J. Slovák, and P. W. Michor. Natural operations in differential geometry [4] J. M. Lee. Smooth manifolds. Springer, [5] J. M. Lee. Riemannian manifolds: an introduction to curvature, volume 176. Springer Science & Business Media, [6] J. M. Lee. Introduction to topological manifolds, volume 940. Springer Science & Business Media, [7] M. Spivak. A comprehensive introduction to differential geometry. Vol. II. Publish or Perish Inc., Wilmington, Del., second edition,