Lecture 11 Last lecture we defined connections on fiber bundles. This lecture we will attempt to explain what a connection does for differential ge- ometers. We start with two basic concepts from algebraic topology. The first is that of homotopy lifting property. Let us recall: Definition 1. Let π : E → B be a continuous map and X a space. We say that (X, π) satisfies homotopy lifting property if for every H : X × I → B and map f : X → E with π ◦ f = H( , 0) there exists a homotopy F : X × I → E starting at f such that π ◦ F = H. If ({∗}, π) satisfies this property then we say that π has the homotopy lifting property for a point. Now, as is well known and easily proven, every fiber bundle satisfies the homotopy lifting property. A much stronger property is the notion of a covering space (which we will not recall with precision). Here we have the homotopy lifting property for a point plus uniqueness or the unique path lifting property. I.e. there is exactly one lift of a given path in the base starting at a specified point in the total space. The point of a connection is to take the ambiguity out of the homotopy lifting property for X = {∗} and force a fiber bundle to satisfy the unique path lifting property. We will see many conceptual advantages of this uniqueness, but let us start by making this a bit clearer via parallel transport. Definition 2. Let B be a topological space and PB the category whose objects are points of B and whose morphisms are defined via {γ : → B : ∃ c, d ∈ s.t. γ(−∞, c) = b , γ(d, ∞) = b } Hom(b , b ) = R R 1 1 1 2 ∼ where γ ∼ γ˜ if one is a compactly supported reparametrization of the other. Composition is defined by the appropriate concatenation. Exercise 11.1. Show that PB is in fact a category. I.e. give a more precise definition of composition and show that it is associative. Let us now define parallel transport. Definition 3. Let π : E → B be a fiber bundle with fiber F and let F be the groupoid of topological spaces diffeomorphic to F . A parallel transport functor P is a functor P : PB → F which on objects takes b to π−1(b). The holonomy group of a par- allel transport functor P at b is the group generated by the image hP (Hom(b, b))i < Aut(Fb). 1 2 While abstract, this definition picks out the relevant aspect of paral- lel transport for a connection. To show that certain nice connections ∇ induce a parallel transport functor P∇, we state a theorem that applies to all connections. Theorem 11.1. Let π : E → B be a fiber bundle with connection η ∈ Ω1(E, V). Assume γ :(−ε, ε) → B with γ(0) = b . Then there is a neighborhood U ⊂ Fb × (−ε, ε) around Fb × {0} such that for every (f, 0) there exists ε ≥ δf > 0 and path Γf (−δf , δf ) → E for which 0 i) Γf (t) ∈ Hη , ii) π ◦ Γf = γ|(−δe,δe) iii) U is the maximal neighborhood for which i and ii hold. ˜ iv) For every reparametrization γ˜ = γ ◦ φ, U and Γf = Γf ◦ φ satisfy properties i, ii and iii. Proof. For the proof of i, ii and iii, we just observe that the vector field ∗ ∗ ∂t on (−ε, ε) lifts to a vector field Lη(∂t) on γ (E) via γ (π). Now i, ii and iii follow from the existence theorem for local solutions to d O.D.E.’s (the differential equation being the one defined by dt f(t) = Lη(∂t)f(t)). For part iv, apply the chain rule to a reparametrization and 0 −1 observe that the resulting vector field to integrate is Lτ (φ (φ (t))∂t) = 0 −1 ∞ φ (φ (t))Lη(∂t) where here we use the C ((−ε, ε)) linearity of L. Thus another application of the chain rule for Γf gives iv). Note that part iii) shows that U is unique. Following Kolar, we say that η is a complete connection if δf = ε for all f ∈ Fb. We proceed now to show the existence of such complete connections. Let us start with a lemma: Lemma 1. Any bounded vector field on a complete manifold is inte- grable for all time. Proof. Assume X is a bounded vector field on M relative to the com- plete metric g. Let γ :(−ε, ε) → M be a maximal integral curve for X with γ(0) = p. Assume ||X||g ≤ B and observe that the length `(γ) of γ is Z ε dγ `(γ) = dt −ε dt g Z ε = ||X||g dt −ε ≤ 2Bε Thus all of the points in the image of γ are at most Bε distance from p. Thus γ is contained in a closed and bounded subset of M. By 3 Hopf-Rinow, γ is contained in a compact subset which implies that limt→ε γ(t) = q exists (it must be unique by the continuity and bounded property of X). Extending the flow locally around q contradicts the maximality of γ. Theorem 11.2. A complete connection exists on every fiber bundle. Proof. We produce a complete metric on the fiber bundle. Let gB be a complete metric on the base and gF one on the fiber. Let U = {Ui}i∈I be a locally finite trivializing open cover of B and λi a partition −1 of unity subordinate to U. We have maps φi : Ui × F → π (Ui) and we can define the symmetric two form (φi)∗((λigB) + gF ) on E. P Summing gives gE = i∈I (φi)∗((λigB) + gF ). Clearly, π then is a Riemannian submersion (i.e. the orthogonal complements of the fibers map isometrically to the base). We claim that gE is complete. This is a consequence of Hopf-Rinow and the following argument. If p1, , p2,... is a Cauchy sequence in E, then by the triangle inequality and definition of gE, the image of this sequence {π(p1), π(p2),...} must converge to q in B. Thus we may restrict to some open neighborhood Ui of q in the cover U for which λi(q) > . One can show that for N >> 0, the distance dgE (pj, pk) ≥ dgF (pj, pk) which implies that the sequence is Cauchy with respect to the fiber direction and metric gF in the −1 trivialization Ui × F = π (Ui). By completeness of this metric, we −1 have that pj → p in π (Ui). Thus gE is complete. Define the connection H = ker(π)⊥. An easy argument using the Riemannian submersion property shows that the lift LH(X) of any bounded vector field X on B will also be bounded. Thus it is integrable for all time and the theorem is established. .