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Affine Connections: Part 2 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/es- tablish 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 1 3rd May 2016 Affine Connections: Part 2 R. Simon Fong 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 2 3rd May 2016 Affine Connections: Part 2 R. Simon Fong 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. ♦ 3 3rd May 2016 Affine Connections: Part 2 R. Simon Fong 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. 4 3rd May 2016 Affine Connections: Part 2 R. Simon Fong 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. 5 3rd May 2016 Affine Connections: Part 2 R. Simon Fong 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 unique- ness 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 6 3rd May 2016 Affine Connections: Part 2 R.
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