70 CHAPTER 3. CONNECTIONS bundle interesting. We prefer to think of sections as \vector valued" maps on M, which can be added and multiplied by scalars, and we'd like to think of the directional derivative ds(x)X as something which respects this linear structure. From this perspective the answer is ambiguous, unless E happens to be the trivial bundle M × Fm ! M. In that case, it makes sense to think Chapter 3 of the section s simply as a map M ! Fm, and the directional derivative is then d s(γ(t)) − s(γ(0)) ds(x)X = s(γ(t)) = lim (3.1) dt t!0 t Connections t=0 for any smooth path with γ_ (0) = X, thus defining a linear map m ds(x) : TxM ! Ex = F : Contents If E ! M is a nontrivial bundle, (3.1) doesn't immediately make sense 3.1 Parallel transport . 69 because s(γ(t)) and s(γ(0)) may be in different fibers and cannot be added. 3.2 Fiber bundles . 72 Yet in this case, we'd like to think of different fibers as being equivalent, so 3.3 Vector bundles . 75 that ds(x) can still be defined as a linear map TxM ! Ex. The problem 3.3.1 Three definitions . 75 is that there is no natural isomorphism between Ex and Ey for x =6 y; we need an extra piece of structure to \connect" these fibers in some way, at 3.3.2 Christoffel symbols . 80 least if x and y are sufficiently close. 3.3.3 Connection 1-forms . 82 This leads to the idea of parallel transport, or parallel translation. The 3.3.4 Linearization of a section at a zero . 84 easiest example to think of is the tangent bundle of a submanifold M ⊂ Rm; R3 3.4 Principal bundles . 88 for simplicity, picture M as a surface embedded in (Figure 3.1). For any vector X 2 TxM at x 2 M and a path γ(t) 2 M with γ(0) = x, it's 3.4.1 Definition . 88 not hard to imagine that the vector X is \pushed" along the path γ in a 3.4.2 Global connection 1-forms . 89 natural way, forming a smooth family of tangent vectors X(t) 2 Tγ(t)M. 3.4.3 Frame bundles and linear connections . 91 In fact, this gives a smooth family of isomorphisms t Pγ : TxM ! Tγ(t)M 0 with Pγ = Id, where \smooth" in this context means that in any local 3.1 The idea of parallel transport t trivialization of T M ! M near x, Pγ is represented by a smooth path of matrices. The reason this is well defined is that M has a natural connec- A connection is essentially a way of identifying the points in nearby fibers tion determined by its embedding in R3; this is known as the Levi-Civit`a of a bundle. One can see the need for such a notion by considering the connection, and is uniquely defined for any manifold with a Riemannian following question: metric (see Chapter 4). Notice that in general if γ(0) = x and γ(1) = y, the isomorphism Given a vector bundle π : E ! M, a section s : M ! E and a vector 1 Pγ : TxM ! TyM may well depend on the path γ, not just its endpoints. X 2 TxM, what is meant by the directional derivative ds(x)X? One can easily see this for the example of M = S2 by starting X as a vector along the equator and translating it along a path that moves first If we regard a section merely as a map between the manifolds M and along the equator, say 90 degrees of longitude, but then makes a sharp E, then one answer to the question is provided by the tangent map T s : turn and moves straight up to the north pole. The resulting vector Y T M ! T E. But this ignores most of the structure that makes a vector at the north pole is different from the vector obtained by transporting X 69 PSfrag replacements 3.1. PARALLEL TRANSPORγ T 71 72 CHAPTER 3. CONNECTIONS X(p0) M p0 PSfrag replacements v0 γ R4 X(p0) C M x y v Ex R4 0 E p0 y C Figure 3.1: Parallel transport of two tangent vectors along a path in a x y surface. Ex Ey along the most direct northward path from x to y. Equivalently, one can Figure 3.2: Parallel transport of a vector along a closed path in S2 ⊂ R3 translate a vector along a closed path and find that it returns to a different leads to a different vector upon return. place in the tangent space than where it began (see Figure 3.2). As we will see in Chapter 5, these are symptoms of the fact that S2 has nontrivial curvature. Once again we are differentiating a path of vectors in the same fiber E , For a general vector bundle π : E ! M, we now wish to associate with x so rs(x)X 2 Ex. We can now begin to deduce what conditions must be any path γ(t) 2 M a smooth family of parallel transport isomorphisms t imposed on the isomorphisms Pγ: first, we must ensure that the expression t Pγ : Eγ(0) ! Eγ(t); (3.3) depends only on s and X = γ_ (0), not on the chosen path γ. Let us 0 2 assume this for the moment. Then for any vector field X 2 Vec(M), the with Pγ = Id. These are far from unique, and as the example of S shows, we must expect that they will depend on more than just the endpoints of covariant derivative defines another section rX s : M ! E. Any sensible the path. But the isomorphisms are also not arbitrary; we now determine use of the word \derivative" should require that the resulting map what conditions are needed to make this a useful concept. rs(x) : TxM ! Ex t The primary utility of the family Pγ is that, once chosen, it enables us to differentiate sections along paths. Namely, suppose γ(t) 2 M is a be linear for all x. This is not automatic; it imposes another nontrivial smooth path through γ(0) = x, and we are given a smooth section along γ, condition on our definition of parallel transport. It turns out that from i.e. a map s(t) with values in E such that π ◦ s(t) = γ(t) (equivalently, this these two requirements, we will be able to deduce the most elegant and is a section of the pullback bundle γ∗E, cf. x2.3). We define the covariant useful definition of a connection for vector bundles. derivative of s along γ, D d − 3.2 Connections on fiber bundles s(t) := r s(0) := (P t) 1 ◦ s(t) : (3.2) dt t dt γ t=0 t=0 − Before doing that, it helps to generalize slightly and consider an arbitrary This is well defined since each of the vectors (P t) 1 ◦ s (t) belongs to the γ fiber bundle π : E ! M, with standard fiber F . Now parallel transport same fiber Ex, thus rts(0) 2 Ex. Defining rts(t) similarly for all t gives along a path γ(t) 2 M will be defined by a smooth family of diffeomor- another smooth section of E along γ. t phisms Pγ : Eγ(0) ! Eγ(t), and we define covariant derivatives again by for- For a smooth section s : M ! E, we are also now in a position to define mulas (3.2) and (3.3). Now however, we are differentiating paths through directional derivatives. Reasoning by analogy, we choose a path γ(t) 2 M the fiber Ex ∼= F , which is generally not a vector space, so rs(x)X is with γ(0) = x and γ_ (t) = X 2 TxM, and define the covariant derivative not in the fiber itself but rather in its tangent space Ts(x)(Ex) ⊂ Ts(x)E. d Remember that the total space E is itself a smooth manifold, and has its rs(x)X := r s := (P t)−1 ◦ s(γ(t)) : (3.3) X dt γ own tangent bundle T E ! E. t=0 3.2. FIBER BUNDLES 73 74 CHAPTER 3. CONNECTIONS Definition 3.1. Let π : E ! M be a fiber bundle. The vertical bundle thus V E ! E is the subbundle of T E ! E defined by d t Horp0 (γ_ (0)) = Pγ(p0) = Y (0; p0) = T s(γ_ (0)) − rγ_ (0)s: (3.5) V E ξ T E π∗ξ : dt = f 2 j = 0g t=0 Its fibers VpE := (V E)p ⊂ TpE are called vertical subspaces. This expression is clearly a linear function of γ_ (0). It is also injective since rγ_ (0)s 2 Vp0 E, and T s(γ_ (0)) 2 Vp0 E if and only if γ_ (0) = 0, as we can see Then VpE = Tp(Eπ p ), so the vertical subbundle is the set of all vectors ( ) by applying π∗. The same argument shows (im Horp0 ) \ Vp0 E = f0g, and in T E that are tangent to any fiber. since any non-vertical vector ξ 2 Tp0 E n Vp0 E can be written as T s(γ_ (0)) for some path γ and section s, clearly Exercise 3.2. Show that V E ! E is a smooth real vector bundle if π : E ! M is a smooth fiber bundle, and the rank of V E ! E is the im Horp ⊕Vp E = Tp E: dimension of the standard fiber F .