18 CHAPTER 2. BUNDLES Example 2.1 (The tangent bundle). If M is a smooth n-dimensional manifold, its tangent bundle T M = TxM x2M [ associates to every x M the n-dimensional vector space TxM. All of Chapter 2 these vector spaces are2 isomorphic (obviously, since they have the same dimension), though it's important to note that there is generally no natural choice of isomorphism between T M and T M for x = y. There is however x y 6 Bundles a sense in which the association of x M with the vector space TxM can 2 be thought of as a \smooth" function of x. We'll be more precise about this later, but intuitively one can imagine M as the sphere S2, embedded smoothly in R3. Then each tangent space is a 2-dimensional subspace of R3, 2 R3 2 Contents and the planes TxS vary smoothly as x varies over S . Of course, the definition of \smoothness"⊂ for a bundle should ideally not depend on any 2.1 Vector bundles . 17 embedding of M in a larger space, just as the smooth manifold structure 2.2 Smoothness . 26 of M is independent of any such embedding. The general definition will 2.3 Operations on vector bundles . 28 have to reflect this. 2.4 Vector bundles with structure . 33 In standard bundle terminology, the tangent bundle is an example of a smooth vector bundle of rank n over M. We call M the base of this bundle, 2.4.1 Complex structures . 33 and the 2n-dimensional manifold T M itself is called its total space. There 2.4.2 Bundle metrics . 35 is a natural projection map π : T M M which, for each x M, sends ! −1 2 2.4.3 Volume forms and volume elements . 37 every vector X TxM to x. The preimages π (x) = TxM are called fibers 2 2.4.4 Indefinite metrics . 43 of the bundle, for reasons that might not be obvious at this stage, but will become more so as we see more examples. 2.4.5 Symplectic structures . 45 Example 2.2 (Trivial bundles). This is the simplest and least interesting 2.4.6 Arbitrary G-structures . 48 example of a vector bundle, but is nonetheless important. For any manifold 2.5 Infinite dimensional bundles . 49 M of dimension n, let E = M Rm for some m N. This is a manifold 2.6 General fiber bundles . 56 of dimension n + m, with a natural× projection map2 2.7 Structure groups . 59 π : M Rm M : (x; v) x: 2.8 Principal bundles . 63 × ! 7! We associate with every point x M the set E := π−1(x) = x Rm, 2 x f g × which is called the fiber over x and carries the structure of a real m- dimensional vector space. The manifolds E and M, together with the projection map π : E M, are collectively called the trivial real vector ! 2.1 Vector bundles: definitions and exam- bundle of rank m over M. Any map s : M E of the form ! ples s(x) = (x; f(x)) Roughly speaking, a smooth vector bundle is a family of vector spaces that is called a section of the bundle π : E M; here f : M Rm is an are \attached" to a manifold in some smoothly varying manner. We will arbitrary map. The terminology comes from! the geometric in!terpretation present this idea more rigorously in Definition 2.8 below. First though, it's of the image s(M) E as a \cross section" of the space M Rm. Sections worth looking at some examples to understand why such an object might can also be characterized⊂ as maps s : M E such that ×π s = Id , a ! ◦ M be of interest. definition which will generalize nicely to other bundles. 17 2.1. VECTOR BUNDLES 19 20 CHAPTER 2. BUNDLES Remark 2.3. A word about notation before we continue: since the essential which consists of multilinear maps information of a vector bundle is contained in the projection map π, one ∗ ∗ R usually denotes a bundle by \π : E M". This notation is convenient TxM : : : TxM Tx M : : : Tx M : ! × × × × × ! because it also mentions the manifolds E and M, the total space and base ` k respectively. When there's no ambiguity, we often omit π itself and denote Thus (T kM) |has dimension{z nk}+`, |and {z } a bundle simply by E M, or even just E. ` x ! T kM M Consider for one more moment the first example, the tangent bundle ` ! π : T M M. A section of the bundle π : T M M is now defined to be is a real vector bundle of rank nk+`. In particular, T 1M = T M has rank n, any map!s : M T M that associates with eac!h x M a vector in the 0 and so does T 0M = T ∗M. For k = ` = 0, we obtain the trivial line bundle fiber T M; equiv!alently, s : M T M is a section if2π s = Id . Thus 1 x ! ◦ M T 0M = M R, whose sections are simply real valued functions on M. For \section of the tangent bundle" is simply another term for a vector field 0 × k 0, T 0M contains an important subbundle on M. We denote the space of vector fields by Vec(M). ≥ k If M is a surface embedded in R3, then it is natural to think of each 0 k ∗ Tk M Λ T M M 3 tangent space TxM as a 2-dimensional subspace of R . This makes π : ⊃ ! R3 n! 1 T M M a smooth subbundle of the trivial bundle M M, in of rank k!(n−k)! , whose sections are the differential k-forms on M. that eac! h fiber is a linear subspace of the corresponding fib×er of the! trivial Notation. In all that follows, we will make occasional use of the Einstein bundle. summation convention, which already made an appearance in the introduc- tion, 1.2. This is a notational shortcut by which a summation is implied Example 2.4 (The cotangent bundle). Associated with the tangent x bundle T M M, there is a \dual bundle" T ∗M M, called the cotan- over any index that appears as a pair of upper and lower indices. So for gent bundle:!its fibers are the vector spaces ! instance if the index j runs from 1 to n, n n ∗ j j j @ j @ T M = HomR(T M; R); λ dx := λ dx ; and X := X ; x j j @xj @xj j=1 j=1 X X consisting of real linear maps TxM R, also known as dual vectors. The j sections of T ∗M are thus precisely!the differential 1-forms on M. This in the latter expression, the index in @x counts as a \lower" index by is our first example of an important vector bundle that cannot easily be convention because it's in the denominator. The summation convention visualized, though one can still imagine that the vector spaces T ∗M vary and the significance of \upper" vs. \lower" indices is discussed more fully smoothly in some sense with respect to x M. in Appendix A. ∗ 2 Observe that T M has the same rank as T M, and indeed there are Example 2.6 (Distributions). Suppose λ is a smooth 1-form on the ∗ always isomorphisms TxM ∼= Tx M for each x, but the choice of such an manifold M which is everywhere nonzero. Then we claim that at every isomorphism is not canonical. This is not a trivial comment: once we define point p M, the kernel of λ TpM is an (n 1)-dimensional subspace ξp what \isomorphism" means for vector bundles, it will turn out that T M 2 j 1 − n ⊂ TpM. To see this, choose coordinates (x ; : : : ; x ) near a point p0 so that ∗ and T M are often not isomorphic as bundles, even though the individual λ can be written as fibers T M and T ∗M always are. j x x λ = λj dx ; Example 2.5 (Tensor bundles). The tangent and cotangent bundles are using the set of real-valued \component functions" λ1(p); : : : ; λn(p) defined n k near p0. By assumption the vector (λ1(p); : : : ; λn(p)) R is never zero, both examples of a more general construction, the tensor bundles T` M 2 M, whose sections are the tensor fields of type (k; `) on M. For integers! and the kernel of λ at p is precisely the set of tangent vectors k k 0 and ` 0, the fiber (T` M)x over x M is defined to be the vector ≥ ≥ 2 j @ space X = X j TpM ` k @x 2 T ∗M T M ; 1The notions of multilinear maps and tensor products used here are discussed in x ⊗ x j=1 ! j=1 ! detail in Appendix A. O O 2.1. VECTOR BUNDLES 21 22 CHAPTER 2. BUNDLES j such that λ(X) = λj(p)X = 0, i.e. it is represented in coordinates by the With these examples in mind, we are ready for the general definition. n orthogonal complement of (λ1(p); : : : ; λn(p)) R , an (n 1)-dimensional The rigorous discussion will begin in a purely topological context, where subspace. 2 − the notion of continuity is well defined but smoothness is not|smooth The union of the subspaces ker λ for all p defines a smooth rank- structures on vector bundles will be introduced in the next section.