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Example 2.1 (The tangent bundle). If M is a smooth n-dimensional manifold, its tangent bundle

T M = TxM x∈M [ associates to every x M the n-dimensional vector space TxM. All of Chapter 2 these vector spaces are∈ 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 ∈ 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 ∈ 2.4.3 Volume forms and volume elements ...... 37 every vector X TxM to x. The preimages π (x) = TxM are called fibers ∈ 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 map∈ 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, ∈ x { } × 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.

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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 if∈π 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 § 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. ∗ ∈ 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 ∈ | 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 ∈ 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 ≥ ≥ ∈ j ∂ space X = X j TpM ` k ∂x ∈ 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. ∈ − 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. |TpM (n 1) subbundle − Notation. In everything that follows, we choose a field ξ M → F R C of the tangent bundle, also known as an (n 1)-dimensional distribution = either or , on M. More generally, one can define smooth−m-dimensional distributions and assume unless otherwise noted that all vector spaces and linear maps for any m n, all of which are rank-m subbundles of T M. In this case, ≤ are F-linear. In this way the real and complex cases can be handled simul- “smoothness” means that in some neighborhood of any point p M, ∈ taneously. one can find a set of smooth vector fields X1, . . . , Xm that are linearly independent and span ξq at every point q close to p. A section of ξ is by Definition 2.8. A vector bundle of rank m consists of a pair of topological definition a vector field X Vec(M) such that X(p) ξp for all p M. spaces E and M, with a continuous surjective map π : E M such that ∈ ∈ ∈ −1 → for each x M, the subset Ex := π (x) E is a vector space isomorphic 3 Example 2.7 (A complex line bundle over S ). All of the examples so to Fm, and∈ for every point x M there⊂ exists an open neighborhood ∈ far have been real vector bundles, but one can just as well define bundles x M and a local trivialization with fibers that are complex vector spaces. For instance, define S3 as the ∈ U ⊂ unit sphere in R4, and identify the latter with C2 via the correspondence Φ : π−1( ) Fm. U → U ×

(x , y , x , y ) (x + iy , x + iy ). Here Φ is a homeomorphism which restricts to a linear isomorphism Ey 1 1 2 2 1 1 2 2 → ←→ y Fm for each y . 2 { } × ∈ U Then if , C2 denotes the standard complex inner product on C , one We call E the total space of the bundle π : E M, and M is called h i 4 −1 → finds that the real inner product , R4 on R is given by the base. For each x M, the set Ex = π (x) E is called the fiber over h i x. ∈ ⊂ X, Y R4 = Re X, Y C2 , h i h i Definition 2.9. A vector bundle of rank 1 is called a line bundle. and in particular X, X R4 = X, X C2 . Thus by this definition, h i h i A bundle of rank m is sometimes also called an m-dimensional vector 3 2 bundle, or an m-plane bundle. The use of the term “line bundle” for m = 1 S = z C z, z C2 = 1 . F R { ∈ | h i } is quite intuitive when = ; one must keep in mind however that in the 3 complex case, the fibers should be visualized as planes rather than lines. The tangent space TzS is then the real orthogonal complement of the Given a vector bundle π : E M and any subset M, denote C2 R4 4 vector z = with respect to the inner product , R . But there is E = π−1( ). As in the definition→ above, a trivializationU ⊂over is a ∈ h i |U U U also a complex orthogonal complement homeomorphism Fm 2 Φ : E U ξz := v C z, v C2 = 0 , | → U × { ∈ | h i } that restricts linearly to the fibers as isomorphisms 3 which is both a real 2-dimensional subspace of TzS and a complex 1- m C2 3 Ex x F dimensional subspace of . The union of these for all z S defines a → { } × vector bundle ξ S3, which can be thought of as either∈a real rank-2 → for each x . One cannot expect such a trivialization to exist for every subbundle of T S3 S3 or as a complex rank-1 subbundle of the trivial ∈ U → subset M; when it does exist, we say that π : E M is trivializable complex bundle S3 C2 S3.2 U ⊂ → × → over . Thus every vector bundle is required to be locally trivializable, in U 2The distribution ξ T S3 is well known in contact geometry as the standard contact the sense that trivializations exist over some neighborhood of each point in structure on S3. ⊂ the base. The bundle is called (globally) trivializable, or simply trivial, if 2.1. VECTOR BUNDLES 23 24 CHAPTER 2. BUNDLES there exists a trivialization over the entirety of M. We’ve already seen the Exercise 2.13. Show that a line bundle π : E M is trivial if and only → archetypal example of a globally trivial bundle: the product E = M Fm. if there exists a continuous section s : M E that is nowhere zero. × → Nontrivial bundles are of course more interesting, and are also in some The example above shows why one often thinks of nontrivial vector sense more common: e.g. the tangent bundle of any closed surface other bundles as being “twisted” in some sense—this word is often used in ge- than a torus is nontrivial, though we’re not yet in a position to prove this. ometry and topology as well as in physics to describe theories that depend Example 2.12 below shows a nontrivial line bundle that is fairly easy to crucially on the nontriviality of some bundle. understand. Given two vector bundles E M and F M, of rank m and ` → → Definition 2.10. A section of the bundle π : E M is a map s : M E respectively, a linear bundle map A : E F is any continuous map that → → restricts to a linear map E F for eac→h x M. A bundle isomorphism such that π s = IdM . x → x ∈ ◦ is a homeomorphism which is also a linear bundle map—in this case, the The space of continuous sections on a vector bundle π : E M is inverse is a linear bundle map as well. Hence E M is trivializable if and → denoted by only if it is isomorphic to a trivial bundle M →Fm M. × → Γ(E) = s C(M, E) π s = IdM . There is such a thing as a nonlinear bundle map as well, also called a { ∈ | ◦ } fiber preserving map. The map f : E F is called fiber preserving if for This is an infinite dimensional vector space, with addition and scalar mul- → all x M, f(E ) F . tiplication defined pointwise. In particular, every vector bundle has a pre- ∈ x ⊂ x ferred section 0 Γ(E), called the zero section, which maps each point Definition 2.14. Let π : E M be a vector bundle. A subbundle of E is ∈ → x M to the zero vector in Ex. This gives a natural embedding M , E. a subset F E such that the restriction π : F M is a vector bundle, ∈ → ⊂ |F → and the inclusion F , E is a continuous linear bundle map. Remark 2.11. As we mentioned already in Example 2.2, the term “section” → refers to the geometric interpretation of the image s(M) as a subset of the Example 2.15. The line bundle of Example 2.12 is a subbundle of the total space E. One can alternately define a continuous section of the trivial 2-plane bundle S1 R2 S1. bundle π : E M to be any subset Σ E such that π Σ : Σ M is a × → homeomorphism.→ ⊂ | → Real vector bundles can be characterized as orientable or non-orientable. Recall that an orientation for a real m-dimensional vector space V is a Example 2.12 (A nontrivial line bundle). Identify S1 with the unit choice of an equivalence class of ordered bases circle in C, and define a bundle ` S1 to be the union of the sets eiθ 1 2 → { } × (v(1), . . . , v(m)) V, `eiθ S R for all θ R, where the fibers `eiθ are the 1-dimensional ⊂ subspaces⊂ × ∈ where two such bases are equivalent if one can be deformed into the other cos(θ/2) 2 through a continuous family of bases. A basis in the chosen equivalence ` iθ = R R . e sin(θ/2) ⊂ class is called a positively oriented basis, while others are called negatively   If we consider the subset oriented. There are always two choices of orientation, owing to the fact that the general linear group GL(m, R) has two connected components, (eiθ, v) ` θ R, v 1 distinguished by the sign of the determinant.3 If V and W are oriented { ∈ | ∈ | | ≤ } vector spaces, then an isomorphism A : V W is called orientation consisting only of vectors of length at most 1, we obtain a M¨obius strip. preserving if it maps every positively oriented→basis of V to a positively Observe that this bundle does not admit any continuous section that is oriented basis of W ; otherwise, it is orientation reversing. For the space nowhere zero. It follows then from Exercise 2.13 below that ` is not globally Rm with its natural orientation given by the standard basis of unit vectors m m trivializable. Local trivializations are, however, easy to construct: e.g. for (e(1), . . . , e(m)), an invertible matrix A : R R is orientation preserving 1 → any point p S , define if and only if det A > 0. ∈ This notion can be extended to a real vector bundle π : E M 1 iθ cos(θ/2) iθ → Φ : ` 1 (S p ) R : e , c (e , c). by choosing orientations on each fiber Ex so that the orientations “vary |S \{p} → \ { } × sin(θ/2) 7→    continuously with x”. More precisely: But this trivialization can never be extended continuously to the point p. 3By contrast, GL(m, C) is connected, which is why orientation makes no sense for (Spend a little time convincing yourself that this is true.) complex vector spaces. 2.1. VECTOR BUNDLES 25 26 CHAPTER 2. BUNDLES

Definition 2.16. An orientation of the real vector bundle π : E M is a Proof. One simply defines Φ(v (x)) = (x, e ) and extends Φ to the rest of → (j) (j) choice of orientation for every fiber Ex, such that for any local trivialization the fiber Ex by linearity: thus using the summation convention to express m m j j Φ : E U R , the induced isomorphisms Ex x R for x arbitrary vectors X = X v(j) Ex in terms of components X F, are all| either→ U orien× tation preserving or orientation rev→ ersing.{ } × ∈ U ∈ ∈ X1 Not every real bundle admits an orientation; those that do are called j . orientable. Φ(X) = (x, X e(j)) = x,  .  . Xm Exercise 2.17. Show that a real line bundle is orientable if and only if it       is trivializable. In particular, the bundle of Example 2.12 is not orientable. The resulting map on E is clearly continuous. |U A smooth manifold M is called orientable if its tangent bundle is ori- Obvious as this result seems, it’s useful for constructing and for visual- entable. This is equivalent to a more general notion of orientation which izing trivializations, since frames are in some ways more intuitive objects. is also defined for topological manifolds, even without a well defined tan- gent bundle. The more general definition is framed in terms of homology theory; see for example [Hat02]. 2.2 Smoothness Example 2.18. Using again the nontrivial line bundle ` S1 of Exam- ple 2.12, let → So far this is all purely topological. To bring things into the realm of Σ = (eiθ, v) ` v 1 . differential geometry, some notion of “differentiability” is needed. { ∈ | | | ≤ } To define this, we note first that every vector bundle admits a system This is a well known manifold with boundary: the M¨obius strip. To see m of local trivializations, i.e. a set of trivializations Φα : E U α F such that it is non-orientable, imagine placing your thumb and index finger | α → U × that the open sets α cover M. Such a system defines a set of continuous on the surface to form an oriented basis of the tangent space at some transition maps {U } point: assume the thumb is the first basis vector and the index finger is gβα : α β GL(m, F), the second. Now imagine moving your hand once around, deforming the U ∩ U → so that each of the maps Φ Φ−1 : ( ) Fm ( ) Fm takes basis continuously as it goes. When it comes back to the same point, you β ◦ α Uα ∩ Uβ × → Uα ∩ Uβ × will have a basis that cannot be deformed back to the original one—you the form (x, v) (x, g (x)v). would have to rotate your hand to recover its original position, pointing in 7→ βα directions not tangent to Σ along the way. Exercise 2.21. Show that a real vector bundle π : E M is orientable if Other popular examples of non-orientable surfaces include the projective and only if it admits a system of local trivializations for→which the transition plane and the Klein bottle, cf. [Spi99]. maps gβα : α β GL(m, R) all satisfy det gβα(x) > 0. Definition 2.19. Let π : E M be a vector bundle of rank m, and U ∩ U → choose a subset M of the →base. A frame Definition 2.22. Suppose M is a smooth manifold and π : E M is a U ⊂ vector bundle. A smooth structure on π : E M is a maximal→ system → (v(1), . . . , v(m)) of local trivializations which are smoothly compatible, meaning that all transition maps gβα : α β GL(m, F) are smooth. over is an ordered set of continuous sections v(j) Γ(E U ), such that for U ∩ U → U ∈ | The bundle π : E M together with a smooth structure is called a every x the set (v(1)(x), . . . , v(m)(x)) forms a basis of Ex. ∈ U smooth vector bundle. → A global frame is a frame over the entire base M; as the next statement shows, this can only exist if the bundle is trivializable. The smoothness of the transition maps gβα : α β GL(m, F) is U ∩ U → F Proposition 2.20. A frame (v , . . . , v ) over M for the bundle judged by choosing local coordinates on M and treating GL(m, ) as an (1) (m) F Fm π : E M determines a trivialization U ⊂ open subset of the vector space of -linear transformations on . → Remark 2.23. One can also define vector bundles of class C k, by requiring Φ : E Fm |U → U × the transition maps to be k-times differentiable. For this to make sense k such that for each x and j 1, . . . , m , Φ(v(j)(x)) = (x, e(j)), where when k 1, the base M must also have the structure of a C -manifold, ∈ U ∈ { } m ≥ 0 (e(1), . . . , e(m)) is the standard basis of unit vectors in F . or better. By this definition, all vector bundles are bundles of class C 2.2. SMOOTHNESS 27 28 CHAPTER 2. BUNDLES

(M need not even be a manifold when k = 0, only a topological space). For bundles of class Ck with k , it makes sense to speak also of ≤ ∞ Almost all statements below that use the word “smooth” can be adapted Cr-sections for any r k; these can be defined analogously to smooth to assume only finitely many derivatives. sections, by either of the≤ two approaches outlined above. The space of sections of class Cr is denoted by Cr(E). Proposition 2.24. If π : E M is a smooth vector bundle of rank m → Similar remarks apply to linear bundle maps between differentiable and dim M = n, then the total space E admits a natural smooth manifold vector bundles E M and F M of rank m and ` respectively. In structure such that π is a smooth map. The dimension of E is n + m if → → F = R, or n + 2m if F = C. particular, a smooth bundle map A : E F can be expressed in local trivializations as a map of the form → Exercise 2.25. Prove Proposition 2.24. Hint: for x and v E , use ∈ Uα ∈ x the trivialization Φα and a coordinate chart for some open neighborhood (x, v) (x, B(x)v) 7→ x α to define a coordinate chart for the open neighborhood v ∈ U ⊂ U ∈ Fm F` E U E. for some smooth function B(x) with values in Hom( , ). | ⊂ Proposition 2.20 can now be ammended to say that any smooth frame The trivial bundle M Rm M obviously has a natural smooth (v , . . . , v ) over an open subset or submanifold M determines a structure if M is a smooth ×manifold.→ We can construct a smooth structure (1) (m) smooth trivialization over . U ⊂ for the tangent bundle T M M as follows. Pick an open covering M = U → Rn We also have an addendum to Definition 2.14: a subbundle F E is α α and a collection of smooth charts ϕα : α Ωα . Then define ⊂ U Rn U → ⊂ called a smooth subbundle if π F : F M admits a smooth structure such local trivializations Φα : T M Uα α by | → S | → U × that the inclusion F , E is a smooth linear bundle map. → Φα(X) = (x, dϕα(x)X), Exercise 2.28. If ξ T M is a smooth distribution on the manifold M, −1 ⊂ for x α and X TxM. Since ϕβ ϕα is always a smooth diffeomorphism show that ξ M is a smooth subbundle of T M M. between∈ Uopen subsets∈ of Rn, the transition◦ maps → → RN −1 Exercise 2.29. If M is a smooth submanifold of , show that T M M gβα(x) = d(ϕβ ϕ )(ϕα(x)) GL(n, R) → ◦ α ∈ is a smooth subbundle of the trivial bundle M RN M. × → are smooth. Many of the statements that follow can be applied in obvious ways to Exercise 2.26. Verify that the tensor bundles and distributions from Ex- either general vector bundles or smooth vector bundles. Since we will need amples 2.5, 2.6 and 2.7 are all smooth vector bundles. to assume smoothness when connections are introduced in Chapter 3, the In the context of smooth vector bundles, it makes sense to require that reader may as well assume for simplicity that from now on all bundles, all sections and bundle maps be smooth. We thus redefine Γ(E) as the sections and bundle maps are smooth—with the understanding that this vector space of all smooth sections, assumption is not always necessary. Γ(E) = s C∞(M, E) π s = Id . { ∈ | ◦ M } Note that this definition requires the smooth manifold structure of the total 2.3 Operations on vector bundles space E, using Proposition 2.24. The following exercise demonstrates an There are many ways in which one or two vector bundles can be combined alternative definition which is also quite revealing, and makes no reference or enhanced to create a new bundle. We now outline the most important to the total space. examples; in all cases it is a straightforward exercise to verify that the new Exercise 2.27. Show that the following notion of a smooth section is objects admit (continuous or smooth) vector bundle structures. equivalent to the definition given above: a smooth section is a map s : m M E such that for every smooth local trivialization Φ : E U F , the→map Φ s : Fm takes the form | → U × Bundles of linear maps ◦ |U U → U × Φ s(x) = (x, f(x)) For two vector spaces V and W , denote by ◦ for some smooth map f : Fm. Hom(V, W ) U → 2.3. OPERATIONS ON VECTOR BUNDLES 29 30 CHAPTER 2. BUNDLES the vector space of linear maps V W . This notation assumes that both The dual bundle → spaces are defined over the same field F, in which case Hom(V, W ) is also ∗ F an F-linear vector space. If one or both spaces are complex, one can always The dual space V of a vector space V is Hom(V, ), and its elements are still define a space of real linear maps called dual vectors. The extension of this concept to bundles is a special case of the definition above: we choose F to be the trivial line bundle HomR(V, W ) M F M, and define the dual bundle of E M by by regarding any complex space as a real space of twice the dimension. × → → Note that HomR(V, W ) is actually a complex vector space if W is complex, E∗ = Hom(E, M F) M. since the scalar multiplication × → ∗ (λA)(v) = λA(v) For each x M, the fiber E = Hom(Ex, F) is the dual space of Ex. ∈ x For example, the dual of a tangent bundle T M M is the cotangent then makes sense for λ C, A HomR(V, W ) and v V . If V and W are ∈ ∈ ∈ bundle T ∗M M, whose sections are the 1-forms on→M. both complex spaces, we can distinguish Hom(V, W ) from HomR(V, W ) → by using HomC(V, W ) to denote the (complex) space of complex linear maps. This notation will be used whenever there is potential confusion. If V = W , we use the notation Direct sums End(V ) = Hom(V, V ), The direct sum of a pair of vector spaces V and W is the vector space with obvious definitions for EndR(V ) and EndC(V ) when V is a complex V W = (v, w) v V, w W vector space. ⊕ { | ∈ ∈ } Now for two vector bundles E M and F M over the same base, we define the bundle of linear maps→ → with addition defined by (v, w) + (v0, w0) = (v + v0, w + w0) and scalar multiplication c(v, w) = (cv, cw). Of course, as a set (and as a group Hom(E, F ) M, → under addition), V W is the same thing as V W . The reason for the ⊕ × whose fibers are Hom(E, F )x = Hom(Ex, Fx). This has a natural smooth redundant notation is that V W gives rise to a natural operation on two bundle structure if both E M and F M are smooth. The sections ⊕ → → vector bundles E M and F M, producing a new bundle over M that A Γ(Hom(E, F )) are precisely the smooth linear bundle maps A : E really is not the Cartesian→ product→ E F . The direct sum bundle F .∈We similarly denote by → × End(E) M E F M → ⊕ → the bundle of linear maps from E to itself. The bundles HomF(E, F ) M F R C → is defined to have fibers (E F )x = Ex Fx for x M. and EndF(E) for = or have natural definitions, relating to the ⊕ ⊕ ∈ discussion above. When V and W are both complex vector spaces, we can regard HomC(V, W ) Exercise 2.30. For two complex vector bundles E M and F M, → → as a complex subspace of HomR(V, W ); indeed, show that HomR(E, F ) = HomC(E, F ) HomC(E, F ). ⊕ HomC(V, W ) = A HomR(V, W ) A(iv) = iA(v) for all v V . { ∈ | ∈ } It’s surprisingly complicated to visualize the total space of E F : no- Another interesting subspace is the space of complex antilinear maps tably, it’s certainly not E F . The latter consists of all pairs (v⊕, w) such × HomC(V, W ) = A HomR(V, W ) A(iv) = iA(v) for all v V . that v Ex and w Fy for any x, y M, whereas (v, w) E F only if { ∈ | − ∈ } ∈ ∈ ∈ ∈ ⊕ Thus for complex vector bundles E M and F M, we can define x = y, i.e. v and w are in fibers over the same point of the base. bundles of complex antilinear maps E→ F and E →E respectively: One can define a bundle whose total space is E F ; in fact, for this we → → no longer need assume that E and F have the same× base. Given E M C C → Hom (E, F ) M and End (E) M. and F N, one can define a Cartesian product bundle E F M N → → → × → × These are subbundles of HomR(E, F ) M and EndR(E) M respec- with fibers (E F ) = E F . For our purposes however, this operation → → × (x,y) x⊕ y tively. is less interesting than the direct sum. 2.3. OPERATIONS ON VECTOR BUNDLES 31 32 CHAPTER 2. BUNDLES

1 0 ∗ 1 Tensor products in particular, E0 = E, E1 = E and E1 = End(E). The tensor product bundle E0 = k E∗ has an important subbundle For vector spaces V and W of dimension m and n respectively, the tensor k ⊗j=1 product V W is a vector space of dimension mn. There are various ways ΛkE∗ M, ⊗ to define this, details of which we leave to Appendix A. For now let us → k ∗ whose fibers Λ Ex are spaces of alternating k-forms on Ex. One can simi- simply recall the most important example of a tensor product that arises k k in differential geometry: larly define an alternating tensor product Λ E E0 , consisting of alternat- ing k-forms on E∗. A somewhat more abstract⊂ (but also mathematically ` k cleaner) definition of ΛkE is given in Appendix A. V k = V ∗ V ` ⊗ i=1 ! j=1 ! Exercise 2.32. Show that a real vector bundle E M of rank m is O O orientable if and only if the line bundle ΛmE∗ M is→trivializable. And k+` → is the n -dimensional vector space of all multilinear maps that this is equivalent to the line bundle ΛmE M being trivializable. → V . . . V V ∗ . . . V ∗ F. × × × × × → Complexification ` k Tensor products|extend{z to bundles} | in{zthe ob}vious way: given E M Just as any complex vector bundle can be treated as a real bundle with → and F M, their tensor product is the bundle twice the rank, any real bundle determines a related complex bundle of the → same rank. For an individual real vector space V , the complexification is E F M defined as ⊗ → VC = V R C, whose fibers are (E F ) = E F . There is potential confusion here if ⊗ x x x C both bundles are complex,⊗ so we⊗sometimes distinguish the bundles where is regarded as a 2-dimensional real vector space. The tensor product is a real vector space with twice the dimension of V ; indeed, any E R F M and E C F M. ξ VC can be written as a sum ⊗ → ⊗ → ∈ The former has fibers that are tensor products of E and F , treating both ξ = (v1 1) + (v2 i) x x ⊗ ⊗ as real vector spaces. for some unique pair of vectors v , v V . More importantly, VC can be 1 2 ∈ Exercise 2.31. If V and W are complex vector spaces of dimension p and regarded as a complex vector space by defining scalar multiplication q respectively, show that dimC(V C W ) = pq, while dimR(V R W ) = 4pq. c(v z) := v (cz) Thus these two spaces cannot be⊗identified with one another.⊗ ⊗ ⊗ for c C. Then dimC VC = dimR V , and any real basis of V becomes also Having defined dual bundles and tensor products, the idea of the tensor ∈ C k a complex basis of V . bundles T` M on a manifold can be extended to arbitrary vector bundles The same trick can be used do complexify a real vector bundle E M, by defining → ` k setting C Ek = E∗ E . EC = E R (M ), ` ⊗ ⊗ × i=1 ! j=1 ! where M C M is the trivial complex line bundle, treated as a real O O × → The fiber (Ek) for x M is then the vector space of all multilinear maps vector bundle of rank 2. ` x ∈ E . . . E E∗ . . . E∗ F. x × × x × x × × x → Pullback bundles ` k Suppose π : E M is a smooth vector bundle and f : N M is a → → A little multilinear algebra| {z(see App} endix| A){z shows} that there is a canonical smooth map, with N a smooth manifold. The bundle on M can then be bundle isomorphism “pulled back” via f to a smooth vector bundle Ek = Hom ` E, k E ; f ∗π : f ∗E N, ` ⊗i=1 ⊗j=1 →
2.4. VECTOR BUNDLES WITH STRUCTURE 33 34 CHAPTER 2. BUNDLES with fibers where m denotes the m-by-m identity matrix. We call J0 the standard ∗ 2m 2m (f E)x = Ef(x). complex structure on R , for the following reason: if R is identified Cm This is often called an induced bundle. with via the correspondence (x , . . . , x , y , . . . , y ) (x + iy , . . . , x + iy ), Exercise 2.33. Verify that f ∗E N admits a smooth vector bundle 1 m 1 m ←→ 1 1 m m → m structure if the map f : N M is smooth. then multiplication by i on C is equivalent to multiplication on the left 2m → by J0 in R . Exercise 2.34. For a smooth manifold M, define the diagonal map ∆ : In the same manner, a choice of complex structure J on the space V of M M M : x (x, x). Show that there is a natural isomorphism → × 7→ dimension 2m gives V the structure of an m-dimensional complex vector ∆∗T (M M) = T M T M. × ⊕ space, with scalar multiplication defined by Example 2.35 (Vector fields along a map). Pullback bundles appear (a + ib)v := (a + bJ)v. naturally in many situations, e.g. in Exercise 2.34. For another example, Exercise 2.36. Show that every complex structure J on R2m is similar consider a smooth 1-parameter family of smooth maps to the standard structure J0, i.e. there exists S GL(2m, R) such that −1 ∈ R2m ft : N M SJS = J0. Hint: construct a basis (v1, . . . , vm, w1, . . . , wm) of with → wk = Jvk for each k. for t ( 1, 1). Differentiating with respect to t at t = 0 gives a smooth ∈ − Consider now a real vector bundle E M of rank 2m. A complex map ξ : N T M, → → structure on E M is defined to be any smooth section J : M End(E) ∂ → 2 → ξ(x) = ft(x) . such that for all x M, [J(x)] = Id . This defines each fiber E as a ∂t ∈ − Ex x t=0 complex vector space of dimension m. Clearly ξ(x) T M, thus ξ is a section of the pullback bundle f ∗T M f0(x) 0 Proposition 2.37. The real bundle E M with complex structure J N. Such a section∈ is also known as a vector field along f . → 0 admits the structure of a smooth complex→vector bundle of rank m. Proof. It suffices if for every x M, we can find a neighborhood x ∈ m ∈ U ⊂ 2.4 Vector bundles with extra structure M and a smooth trivialization Φ : E U C which is complex linear | → U × on each fiber, i.e. for y and v Ey, In this section we survey various types of structure that can be added to a ∈ U ∈ Φ(Jv) = iΦ(v). vector bundle, such as metrics, volume elements and symplectic forms. The general pattern is that each additional structure allows us to restrict our This can be done by constructing a smooth family of complex bases for the attention to a system of transition maps that take values in some proper fibers in a neighborhood of x, then applying Proposition 2.20. We leave subgroup of the general linear group. This observation will be important the rest as an exercise—the idea is essentially to solve Exercise 2.36, not when we relate these structures to principal bundles in Section 2.8. just for a single matrix J but for a smooth family of them. This result can be rephrased as follows: since π : E M is a smooth 2.4.1 Complex structures real vector bundle, there is already a maximal system of lo→cal trivializations Φ : E R2m with smooth transition maps As has been mentioned already, a complex vector bundle can always be α |Uα → Uα × g : GL(2m, R). thought of as a real bundle with one additional piece of structure, namely, βα Uα ∩ Uβ → m 2m a definition of “scalar multiplication by i”. To formalize this, consider Identifying C with R such that i = J0, we can identify GL(m, C) with first a real vector space V , with dim V = 2m. A complex structure on the subgroup 2 V is a linear map J End(V ) such that J = , where denotes the C R ∈ − GL(m, ) = A GL(2m, ) AJ0 = J0A . (2.1) identity transformation on V . Note that no such J can exist if V has odd { ∈ | } dimension. (Prove it!) But there are many such structures on R2m, for Then Proposition 2.37 says that after throwing out enough of the triv- example the matrix ializations in the system α, Φα , we obtain a system which still cov- {U } 0 m ers M and such that all transition maps take values in the subgroup J = , 0 −0 GL(m, C) GL(2m, R).  m  ⊂ 2.4. VECTOR BUNDLES WITH STRUCTURE 35 36 CHAPTER 2. BUNDLES

2.4.2 Bundle metrics Example 2.40 (Quantum wave functions). An example of a Hermi- tian vector bundle comes from quantum mechanics, where the probability A metric on a vector bundle is a choice of smoothly varying inner products distribution of a single particle (without spin) in three-dimensional space on the fibers. To be more precise, if π : E M is a smooth vector bundle, → can be modeled by a wave function then a bundle metric is a smooth map ψ : R3 C. , : E E F → h i ⊕ → The probability of finding the particle within a region R3 is given by U ⊂ such that the restriction to each fiber Ex Ex defines a (real or complex) inner product on E . In the real case, ⊕, is therefore a bilinear P (ψ) = ψ(x), ψ(x) dx, x h· ·i|Ex⊕Ex U h i form which is symmetric and positive definite. This is called a Euclidean ZU structure, making the pair (E, , ) into a Euclidean vector bundle. where v, w := v¯w defines the standard inner product on C. Though h i h i In the complex case, the form , is positive definite and sesquilin- one wouldn’t phrase things this way in an introductory class on quantum h· ·i|Ex⊕Ex ear, which means that for all v, w Ex and c C, mechanics, one can think of ψ as a section of the trivial Hermitian line ∈ ∈ bundle (R3 C, , ). This is not merely something a mathematician would (i) v, w = w, v , × h i h i h i say to intimidate a physicist—in quantum field theory it becomes quite (ii) cv, w = c¯ v, w and v, cw = c v, w , important to consider fields as sections of (possibly nontrivial) Hermitian h i h i h i h i (iii) v + v , w = v , w + v , w and v, w + w = v, w + v, w , vector bundles. h 1 2 i h 1 i h 2 i h 1 2i h 1i h 2i (iv) v, v > 0 if v = 0. Where general vector bundles are concerned, the most important the- h i 6 oretical result about bundle metrics is the following. Recall that for two The metric , is then called a Hermitian structure, and (E, , ) is a inner product spaces (V, , ) and (W, , ), an isometry A : V W is h i h i h i1 h i2 → Hermitian vector bundle. an isomorphism such that Proposition 2.38. Every smooth vector bundle π : E M admits a Av, Aw = v, w → 2 1 bundle metric. h i h i for all v, w V . We leave the proof as an exercise—the key is that since metrics are ∈ Proposition 2.41. Let π : E M be a vector bundle with a metric , . positive definite, any sum of metrics defines another metric. Thus we can Then for every x M, there →is an open neighborhood x M andh ia define metrics in local trivializations and add them together via a partition trivialization ∈ ∈ U ⊂ of unity. This is a standard argument in Riemannian geometry, see [Spi99] Φ : E Fm for details. |U → U × such that the resulting linear maps E x Fm are all isometries (with x → { } × Example 2.39 (Riemannian manifolds). The most important applica- respect to , and the standard inner product on Fm). tion of bundle metrics is to the tangent bundle T M M over a smooth h i → manifold. A metric on T M is usually denoted by a smooth symmetric Proof. Using Proposition 2.20, it suffices to construct a frame (v1, . . . , vm) 0 on some neighborhood of x so that the resulting bases on each fiber are tensor field g Γ(T2 M), and the pair (M, g) is called a Riemannian manifold. The ∈metric g determines a notion of length for smooth paths orthonormal. One can do this by starting from an arbitrary frame and γ : [t , t ] M by setting applying the Gram-Schmidt orthogonalization process. 0 1 → t1 A trivialization that defines isometries on the fibers is called an or- length(γ) = γ˙ (t) dt, thogonal trivialization, or unitary in the complex case. As a result of | |g Zt0 Proposition 2.41, we can assume without loss of generality that a sys- tem of local trivializations α, Φα for a Euclidean vector bundle con- where by definition, X g = g(X, X). The study of smooth manifolds {U } with metrics and their| geometric| implications falls under the heading of sists only of orthogonal trivializations, in which case the transition maps p g : GL(m, R) actually take values in the orthogonal group Riemannian geometry. It is a very large subject—we’ll touch upon a few of βα Uα ∩ Uβ → the fundamental principles in our later study of connections and curvature. O(m) = A GL(m, R) ATA = . { ∈ | } 2.4. VECTOR BUNDLES WITH STRUCTURE 37 38 CHAPTER 2. BUNDLES

For a Hermitian vector bundle, instead of O(m) we have the unitary group (b) Show that for any vector bundle π : E M of rank m and any → integer k m, there is a canonical bundle isomorphism Λk(E∗) =

U(m) = A GL(m, C) A†A = , (ΛkE)∗. ≤ { ∈ | } where by definition A† is the complex conjugate of AT. Taking this dis- (c) Conclude, as in the discussion above, that a choice of volume form for cussion a step further, we could formulate an alternative definition of a a real bundle π : E M of rank m defines naturally an isomorphism → Euclidean or Hermitian structure on a vector bundle as a maximal system ΛmE = Λm(E∗), and thus a corresponding volume form for the dual of local trivializations such that all transition maps take values in the sub- bundle E∗ M. → groups O(m) GL(m, R) or U(m) GL(m, C). It’s easy to see that such a system alwa⊂ys defines a unique bundle⊂ metric. These remarks will be of You should take a moment to convince yourself that this defines some- fundamental importance later when we study structure groups. thing consistent with the intuitive notion of volume: in particular, multi- plying any of the vectors vj by a positive number changes the volume by the same factor, and any linear dependence among the vectors (v1, . . . , vm) 2.4.3 Volume forms and volume elements produces a “degenerate” parallelopiped, with zero volume. The standard volume form on Rm is defined so that the unit cube has volume one. Some- On any real vector space V of dimension m, a volume form µ can be defined what less intuitively perhaps, Vol(v , . . . , v ) is not always positive: it as a nonzero element of the one-dimensional vector space ΛmV , called the 1 m changes sign if we reverse the order of any two vectors, since this changes “top-dimensional exterior product”. For example, Rm has a natural volume the orientation of the parallelopiped. Thus one must generally take the form absolute value Vol(v , . . . , vm) to find the geometric volume. µ = e . . . e , | 1 | (1) ∧ ∧ (m) These remarks extend easily to a complex vector space of dimension m, m where (e(1), . . . , e(m)) is the standard basis of unit vectors in R . A choice of though in this case volume can be a complex number—we will forego any volume form µ ΛmV determines a notion of “signed volume” for subsets attempt to interpret this geometrically. In either the real or complex case, ∈ of V as follows: given any ordered m-tuple of vectors (v1, . . . , vm), we define a linear isomorphism the oriented parallelopiped spanned by these vectors to have signed volume A : (V, µ1) (W, µ2) → Vol(v , . . . , v ) R, where 1 m ∈ between m-dimensional vector spaces with volume forms is said to be vol- ume preserving if every parallelopiped P V has the same (signed or v1 . . . vm = Vol(v1, . . . , vm) µ. ⊂ ∧ ∧ · complex) volume as A(P ) W . For example, a matrix A GL(m, F) is ⊂ ∈ volume preserving with respect to the natural volume form on Fm if and Here we’re using the fact that dim ΛmV = 1 and µ = 0 to deduce that only if det A = 1. there always is such a number. If you’re not used to thinking6 about wedge products of vectors but are familiar with differential forms, then you can Exercise 2.43. A linear map A : V W defines a corresponding linear m ∗ think of volume forms alternatively as follows: Λ V is also 1-dimensional, map → m m ∗ thus the choice µ Λ V defines a unique alternating m-form ω Λ V m m A∗ : Λ V Λ W, such that ∈ ∈ → sending v . . . v to Av . . . Av , as well as a pullback ω(v1, . . . , vm) = Vol(v1, . . . , vm). 1 ∧ ∧ m 1 ∧ ∧ m A volume form is therefore equivalent to a choice of a nonzero alternating A∗ : ΛmW ∗ ΛmV ∗, m-form ω on V , and almost all statements in the following about µ ΛmV → m ∗ ∈ ∗ can be rephrased as statements about ω Λ V . such that A ω(v1, . . . , vm) = ω(Av1, . . . , Avm). Show that A : (V, µ1) ∈ → (W, µ2) is volume preserving if and only if A∗µ1 = µ2. Equivalently, defin- m ∗ m ∗ Exercise 2.42. ing volume via m-forms ω1 Λ V and ω2 Λ W , A is volume preserv- ∗ ∈ ∈ ing if and only if A ω2 = ω1. (a) Show that if π : E M is any (real or complex) line bundle, then a choice of smooth no→where zero section s Γ(E) determines a bundle For any vector bundle π : E M of rank m, the top-dimensional ∈ → isomorphism E = E∗ in a natural way. exterior bundles ΛmE M and ΛmE∗ M are line bundles. In the → → 2.4. VECTOR BUNDLES WITH STRUCTURE 39 40 CHAPTER 2. BUNDLES real case, we saw in Exercise 2.32 that these line bundles are trivial if and Note that SO(m) can alternatively be defined as the subgroup only if E M is orientable, in which case there exist smooth sections → m m ∗ SO(m) = A O(m) det A = 1 . µ : M Λ E and ω : M Λ E that are nowhere zero. Such a { ∈ | } section is→very far from unique,→since we can always obtain another via This is equivalent to the fact that every transformation of Rm that preserves multiplication with a smooth positive function f : M (0, ). both the inner product and the orientation also preserves signed volume, We shall call a nowhere zero section µ Γ(ΛmE)→or ω ∞ Γ(ΛmE∗) a ∈ ∈ i.e. a metric and orientation together determine a preferred volume form. volume form on E, because it defines a smoothly varying notion of signed volume Vol( , . . . , ) on each fiber. This should be mostly familiar from Exercise 2.46. Consider Fm with its standard inner product, and denote the special case· where· E is the tangent bundle of a smooth n-manifold the standard unit basis vectors by e(1), . . . , e(m). M; then a volume form is defined to be a nowhere zero differential n-form n ∗ Fm (cf. [Spi65]), i.e. a section of Λ T M. (a) Show that if (v(1), . . . , v(m)) is any other ordered basis of then Even though orientation doesn’t make sense in the complex case, the v . . . v = det(V ) (e . . . e ) above remarks can still be extended almost verbatim to a complex bundle (1) ∧ ∧ (m) · (1) ∧ ∧ (m) m π : E M of rank m if Λ E is trivial. A volume form is again defined as a F Fm → where V GL(m, ) is the matrix whose jth column is v(j) . smooth nowhere zero section µ Γ(ΛmE), and it now defines a “complex ∈ ∈ ∈ Hint: since ΛmFm is 1-dimensional, the expression volume” Vol(v1, . . . , vm) for any ordered m-tuple of vectors in the same fiber. v(1) . . . v(m) (v , . . . , v ) ∧ ∧ (1) (m) 7→ e . . . e Proposition 2.44. Let π : E M be a vector bundle of rank m with (1) ∧ ∧ (m) → a volume form µ Γ(ΛmE). Then for every x M, there is an open defines an antisymmetric m-linear form on Fm. How many such forms ∈ ∈ neighborhood x M and a trivialization are there? ∈ U ⊂ Φ : E Fm (b) If F = R and we give Rm the standard orientation, show that for |U → U × every positively oriented orthonormal basis (v(1), . . . , v(m)), such that the resulting linear maps E x Fm are all volume preserv- x → { } × ing. v . . . v = e . . . e . (1) ∧ ∧ (m) (1) ∧ ∧ (m) Exercise 2.45. Prove Proposition 2.44. Hint: given any frame (v , . . . , v ) 1 m (c) Conclude that every m-dimensional real vector space with an ori- over a subset M, one can multiply v by a smooth function such that 1 entation and an inner product admits a volume form for which all Vol(v , . . . , v U) ⊂1. 1 m ≡ positively oriented unit cubes have signed volume one. In particular, a maximal system of local trivializations , Φ can α α The situation for a complex bundle π : E M with Hermitian struc- be reduced to a system of volume preserving trivializations, {Ufor whic}h all ture , is slightly different. Now a volume form→µ Γ(ΛmE) can be called transition maps take values in the special linear group comphatiblei with , if every fiber has an orthonormal∈ basis with volume h i iθ SL(m, F) = A GL(m, F) det A = 1 . one. Notice however that if Vol(v1, . . . , vm) = 1, then (e v1, v2, . . . , vm) is iθ iθ { ∈ | } another orthonormal basis and Vol(e v1, v2, . . . , vm) = e , so we cannot A Euclidean structure , on an oriented real bundle π : E M require that all orthonormal bases have volume one. Relatedly, there is no determines a unique preferredh violume form µ such that the oriented→cube well defined notion of “orientation” that could help us choose a compatible spanned by any positively oriented orthonormal basis of Ex has volume one. volume form uniquely; given any compatible form µ, another such form is That this is well defined follows from Exercise 2.46 below. Now orientation defined by fµ if f : M C is any smooth function with f = 1. → | | preserving orthogonal transformations are also volume preserving, so we Given , and a compatible volume form µ, an easy variation on h i can combine Propositions 2.41 and 2.44 to assume that all transition maps Proposition 2.44 shows that we can assume all transition maps take values take values in the special orthogonal group in the special unitary group

SO(m) = O(m) SL(m, R). SU(m) = U(m) SL(m, C). ∩ ∩ 2.4. VECTOR BUNDLES WITH STRUCTURE 41 42 CHAPTER 2. BUNDLES

The nonuniqueness of the volume form is related to the fact that SU(m) on some open set M. Since the question is purely local, we lose no U ⊂ is a submanifold of one dimension less in U(m); by contrast, SO(m) and generality in assuming that E is an oriented bundle. The frame |U → U O(m) have the same dimension, the former being a connected component (e(1), . . . , e(m)) is then assumed to be positively oriented, but is otherwise of the latter. arbitrary; in particular, we do not require it to be orthonormal. The metric A non-orientable real bundle π : E M of rank m does not admit now defines a real m-by-m symmetric matrix G with entries any volume form, but one can still define→ a smoothly varying unsigned G = e , e , volume on the fibers. This is most easily seen in the case where there is ij h (i) (j)i a bundle metric , : then it is natural to say that all orthonormal bases h i and using the frame to express vectors v = vie E for x , we have span non-oriented cubes of volume one, and define a positive volume for (i) ∈ x ∈ U all other bases and parallelopipeds accordingly. The resulting structure v, w = G viwj. is called a volume element on the bundle π : E M. In particular, h i ij every Riemannian manifold (M, g), orientable or not,→has a natural volume Now choose an orthonormal frame (eˆ(1), . . . , eˆ(m)); the corresponding ma- element on its tangent bundle, which one can use to define integration of trix G is then the identity, with G = δ . We can use this new frame to real-valued functions on M. ij ij identify E with the trivial Euclidean vector bundle Rm , More generally, one can define a volume element on any real bundle |U → U U × → U thus bconsidering the other frame b(e , . . . , e ) to consist of smooth Rm- π : E M of rank m to be a choice of two nonzero volume forms µ (1) (m) x valued functions on . The vectors (eˆ , . . . , eˆ ) are now simply the ΛmE →on each fiber E , which vary smoothly in the following sense:every∈ (1) (m) x x standard unit basis vectorsU of Rm. By Exercise 2.46, we have x0 M is contained in an open neighborhood M which admits a ∈ m U ⊂ smooth volume form µU : Λ E such that e . . . e = det(V ) eˆ . . . eˆ = det(V )µ (2.2) U → (1) ∧ ∧ (m) (1) ∧ ∧ (m) µ = µ (x)  x  U where µ is the natural volume form determined by the metric and orienta- tion, and V is the matrix whose columns are the vectors e , . . . , e Rm. for all x . The existence of volume elements in general follows from the (1) (m) We can relate the matrices V and G as follows. Computing in the orthonor-∈ existence∈ofU bundle metrics, with a little help from Exercise 2.46. Then mal coordinates, the volume of parallelopipeds is defined by

k ` T k ` T T Gij = e(i), e(j) = Gk`V i V j = (V )i δk`V i = (V V )ij = (V V )ij, v1 . . . vm = Vol(v1, . . . , vm) ( µx) h i ∧ ∧ ·  thus G = V TV , and det G = (det V )2. Combining this with (2.2) gives a for any unordered basis v , . . . , v of E , with the plus or minus sign b 1 m x formula for the volume form in terms of the metric and the frame, chosen so that volume is {positive. } 1 Exercise 2.47. Show that every real bundle with a volume element admits µ = e . . . e . (2.3) √ (1) ∧ ∧ (m) a system of local trivializations such that the transition maps take values det G in the group This assumes there is an orientation and (e(1), . . . , e(m)) is positively ori- A GL(m, R) det A = 1 . { ∈ |  } ented, but it clearly applies to the non-oriented case as well if we insert signs in the appropriate places. Example 2.48 (Volume elements in coordinates). It’s useful to have  Let us apply this formula to the case of a Riemannian manifold (M, g) explicit coordinate expressions for volume elements on Riemannian mani- 1 n folds, particularly if one wants to understand the physics literature, where with coordinates (x , . . . , x ) on some neighborhood M. The coordi- nates determine a framing (∂ , . . . , ∂ ) of T M over U, ⊂and one expresses coordinates are used in preference to geometrically invariant objects.4 We 1 n U begin by finding an expression for the volume form on a general real bundle the metric in these coordinates by the n-by-n symmetric matrix of rank m with metric , , with respect to a local frame (e(1), . . . , e(m)) h i gij = g(∂i, ∂j), 4For instance, physicists tend to define a tensor purely in terms of its coordinate i i j expression, stipulating that it must “transform properly” with respect to coordinate so that if tangent vectors are written X = X ∂i, then g(X, Y ) = gijX Y . changes. A volume element on (M, g) is actually a volume element on the cotangent 2.4. VECTOR BUNDLES WITH STRUCTURE 43 44 CHAPTER 2. BUNDLES bundle T ∗M, and we want to express the natural volume element deter- A form of this type, having three negative eigenvalues and one positive, is mined by g in terms of the frame dx1 . . . dxn on ΛnT ∗M . Assign said to have Lorentz signature—these are important in Einstein’s theory ∧ ∧ |U to the orientation determined by the coordinates, and let µg denote the of relativity. In fact, general relativity deals exclusively with smooth 4- correspU onding volume form on T M . By (2.3) we have manifolds that have Lorentzian metrics.5 |U → U Given a real m-dimensional vector space V with an indefinite (i.e. sym- 1 µg = ∂ . . . ∂n. metric and nondegenerate) inner product , , the index of , is the √det g 1 ∧ ∧ h i h i largest dimension of any subspace W V on which , is negative ⊂ h i|W This is related to the differential n-form Vol( , . . . , ) by definite. Thus the index is at most m, and is zero if and only if the inner · · product is positive definite. Choosing a basis to identify V with Rm, the 1 X . . . Xn = Vol(X , . . . , Xn) ∂ . . . ∂n, symmetric form , can be written as 1 ∧ ∧ 1 √det g 1 ∧ ∧ h i v, w = vTΣw from which we compute h i for some symmetric matrix Σ. Then by a standard result in linear algebra, X1 . . . Xn T Vol(X1, . . . , Xn) = det g ∧ ∧ . Σ = S ΛS for some S O(m) and a diagonal matrix Λ, thus we can ∂1 . . . ∂n change the basis and assume∈ Σ is diagonal without loss of generality. By the p ∧ ∧ The fraction on the right defines an antisymmetric n-linear form on the nondegeneracy condition, Σ must be invertible, so all its diagonal entries 1 n vectors (X1, . . . , Xn), which we see is identical to dx . . . dx . We are are nonzero. We can then rescale the basis vectors and assume finally that ∧ ∧ Σ takes the form thus lead to the formula m−k Vol = det g dx1 . . . dxn. (2.4) ηm,k := ∧ ∧ k

 −  Again, this assumes that pM has an orientation and the coordinates are for some k 0, . . . , m , where k denotes the k-by-k identity matrix. U ⊂ Clearly k is ∈the{index of} , . positively oriented, but we can easily remove this assumption. The proper h i Similar remarks apply to any smooth indefinite metric , on a real expression for a volume element, expressed in arbitrary coordinates, is then h i vector bundle π : E M of rank m. On any fiber Ex, the diagonaliza- 1 n → d volg = det g dx . . . dx , tion procedure outlined above implies that there is an orthonormal basis (v , . . . , v ) of E , which means in this case v , v = δ . where the wedges have been remop ved to indicate that we are no longer 1 m x h i ji  ij keeping track of signs. This shows how to use local coordinate patches Exercise 2.49. Given an orthonormal basis on the fiber Ex, show that it for computing integrals M f d volg of smooth functions on Riemannian can be extended to a smooth orthonormal frame over some open neighbor- manifolds. hood of x. Hint: Gram-Schmidt! R We mention two important consequences of Exercise 2.49: 2.4.4 Indefinite metrics 1. The index of , is locally constant—it is thus uniquely defined over h i It is sometimes useful to consider “metrics” that are not positive definite. every connected component of the base M. On a real vector bundle E M, one can define an indefinite metric to → R 2. The orthonormal frames give rise to a system of local trivializations be a smooth map , : E E which restricts on each fiber to a which are orthogonal in a generalized sense. We can then assume bilinear form that his symmetrici ⊕ and→ nondegenerate; the latter condition that the transition maps take values in the subgroup means there is no nonzero v Ex with v, w = 0 for all w Ex. An ∈ 4 h i ∈ T example of such a bilinear form on R is the Minkowski metric, defined by O(m k, k) = A GL(m, R) A ηm,kA = ηm,k . T − { ∈ | } v, w = v ηw, where η is the symmetric matrix 5 h i The physical significance of Lorentz signature is that space-time is a 4-manifold 1 0 0 0 in which one of the dimensions is not like the other three. Objects with mass are allowed to move through space-time only along so-called time-like paths x(t), which 0 1 0 0 η =  −  . satisfy x˙(t), x˙(t) > 0. This means that in any coordinate system, the motion defined 0 0 1 0 by x(t)his slower thani a certain limiting speed—the speed of light! One can also make a − 0 0 0 1 distinction between space-time directions that point “forward” or “backward” in time.  −    2.4. VECTOR BUNDLES WITH STRUCTURE 45 46 CHAPTER 2. BUNDLES

In the case where E M is oriented, this can further be reduced to for all v, w V . By constructing an appropriate basis (a symplectic basis), → ∈ one can show that every symplectic vector space (V, ω) of dimension 2m is SO(m k, k) = O(m k, k) SL(m, R). symplectically isomorphic to (R2m, ω ). We denote by Sp(m) GL(2m, R) − − ∩ 0 the subgroup consisting of invertible symplectic linear maps.⊂ One can now define a symplectic structure on a vector bundle π : E It should be noted that more refined notions of orientation are possible → 4 M of rank 2m as a smooth family of symplectic structures on the fibers, if k and m k are both nonzero. For instance, Minkowski space (R , η4,3) can be given− separate orientations for the time-like directions and space- i.e. a smooth map ω : E E R like hypersurfaces, and the associated group O(1, 3) (also known as the ⊕ → full Lorentz group) has four components rather than two. In particular, that defines a symplectic structure on each fiber. The pair (E, ω) is then the components of O(1, 3) are distinguished by whether their elements called a symplectic vector bundle, and we have the following statement preserve or reverse the direction of time and the orientation of space. The about local trivializations: two components that either preserve or reverse both make up SO(1, 3), and Proposition 2.51. the subgroup SO(1, 3)+ SO(1, 3) that preserves both is called the proper If (E, ω) is a symplectic vector bundle over M, then for ⊂ every x M, there is an open neighborhood x M and a symplectic orthochronous Lorentz group. (See for example [SU01].) ∈ ∈ U ⊂ For metrics on the tangent bundle of a smooth manifold, most of the trivialization, i.e. a trivialization basic theory of Riemannian geometry (connections, curvature, geodesics Φ : E R2m etc.) applies just as well to the indefinite case. On the other hand the |U → U × existence result, Proposition 2.38, does not apply to nonpositive metrics 2m such that the resulting linear maps Ex x R are all symplectic 2 in general. For example one can use covering spaces to show that S does → { } × 2m (with respect to ω and the standard symplectic structure ω0 on R ). not admit any metric of index 1 (see [Spi99, Problem 9.7]). The proof follows the usual recipe: one needs to construct a symplectic Exercise 2.50. Adapt the arguments of Example 2.48 to write down the frame in a neighborhood of x, which is no more difficult in principle than natural volume element determined by an indefinite metric. One thing to constructing a symplectic basis on a single vector space. We leave the beware of here is that det g may be negative; but this can be fixed by details as an exercise. inserting the appropriate sign changes. Adapting the remarks following Proposition 2.41 to this setting, a sym- plectic vector bundle can alternatively be defined as a bundle with a system 2.4.5 Symplectic structures of local trivializations for which the transition maps take values in Sp(m). Note that every symplectic bundle (E, ω) of rank m inherits a natural On a real vector space V of even dimension 2m, a symplectic structure is volume form, defined on the dual bundle E∗ by an antisymmetric bilinear form ωm = ω . . . ω. ∧ ∧ ω : V V R ⊕ → That this is nonzero follows from the nondegeneracy of ω. Thus symplectic which is nondegenerate, in the sense that ω(v, w) = 0 for all w V only bundles have natural orientations and volume elements; this is related to ∈ the fact that every symplectic matrix A Sp(m) has determinant one, if v = 0. Note that by a simple argument in linear algebra, there is no ∈ i.e. Sp(m) SL(2m, R). symplectic structure on an odd-dimensional vector space (Prove it!). The ⊂ R2m standard symplectic structure on is defined as Example 2.52 (Hermitian bundles as symplectic bundles). A Her- mitian bundle (E, , ) of rank m can be regarded as a real bundle of rank T h i ω0(v, w) = v J0w, 2m, with a complex structure J and a complex-valued bilinear form where J0 is the standard complex structure. For two symplectic vector v, w = g(v, w) + iω(v, w), h i spaces (V, ω1) and (W, ω2), a linear map A : V W is called symplectic if → where g and ω are real bilinear forms. It is easy to check that g defines ω2(Av, Aw) = ω1(v, w) a Euclidean structure on the real bundle E, while ω defines a symplectic 2.4. VECTOR BUNDLES WITH STRUCTURE 47 48 CHAPTER 2. BUNDLES structure. Thus a unitary trivialization on some subset M is also both One then considers solutions x : R M to the differential equation U ⊂ → an orthogonal trivialization and a symplectic trivialization. This is related x˙(t) = XH (x(t)). This may seem rather arbitrary at first, but any physicist to the convenient fact that if GL(m, C) is regarded as the subgroup of will recognize the form that x˙ = XH (x) takes in the special coordinates matrices in GL(2m, R) that commute with the standard complex structure provided by Darboux’s theorem: writing J0, then U(m) = O(2m) Sp(m). x(t) = (q1(t), . . . , qn(t), p1(t), . . . , pn(t)), ∩ Conversely, every symplectic bundle (E, ω) can be given a compatible we find that for k 1, . . . , n , complex structure J, such that g := ω( , J ) defines a Euclidean metric ∈ { } and , := g + iω becomes a Hermitian· metric.· (See [HZ94] for a proof ∂H ∂H h· ·i q˙k = and p˙k = . of this fact.) Thus there is a close correspondence between symplectic and ∂pk −∂qk complex bundles. These are Hamilton’s equations from classical mechanics, which determine Example 2.53 (Symplectic manifolds). Just as in Riemannian geome- the motion of a system with n degrees of freedom in phase space. Thus the try, the most important examples of symplectic vector bundles are tangent symplectic form ω is precisely the structure needed for defining Hamilton’s bundles, though here it is appropriate to add one extra condition. If ω is a equations in a geometrically invariant way on M. symplectic structure on T M for some smooth 2n-dimensional manifold M, We refer to the book by Arnold [Arn89] for further details. then the pair (M, ω) is called an almost symplectic manifold. The condi- tion for removing the word “almost” comes from observing that ω is now a 2.4.6 Arbitrary G-structures differential 2-form on M; we call it a symplectic form, and the pair (M, ω) a symplectic manifold, if ω satisfies the “integrability condition” One thing all of the examples above have in common is that each structure defines a system of preferred local trivializations (e.g. the orthogonal, or dω = 0. symplectic trivializations), with transition maps that take values in some proper subgroup G GL(m, F). We now formalize this idea. The importance of this condition lies largely in the following result, which ⊂ shows that all symplectic manifolds are locally the same. Definition 2.55. Let π : E M be a vector bundle of rank m, and → Proposition 2.54 (Darboux’s theorem). If (M, ω) is a symplectic mani- suppose G is a subgroup of GL(m, F). A G-structure on E is a maximal m fold and x M, then some neighborhood of x admits a coordinate system system of smooth local trivializations Φα : E U α F which cover ∈ | α → U × (q1, . . . , qn, p1, . . . , pn) in which ω takes the canonical form M and have smooth G-valued transition maps: n gβα : α β G. ω = dpk dqk. U ∩ U → ∧ k=1 X The trivializations ( α, Φα) are called G-compatible. U See [Arn89] or [HZ94] for a proof. There is no such statement for Rie- mannian manifolds, where the curvature defined by a metric g provides an This definition is rather hard to use in practice, because one seldom obstruction for different manifolds to be locally isometric. Thus symplectic wants to construct a system of local trivializations explicitly. But what geometry is quite a different subject, and all differences between symplectic it lacks in practicality, it makes up for in striking generality: all of the manifolds manifest themselves on the global rather than local level—this structures we’ve considered so far are special cases of G-structures. On is the subject of the field known as symplectic topology. a real vector bundle for instance, bundle metrics are equivalent to O(m)- R Some motivation for the study of symplectic manifolds comes from the structures, volume forms are SL(m, )-structures and symplectic struc- fact that they are the natural geometric setting for studying Hamiltonian tures are Sp(m)-structures. These particular examples can all be expressed systems. Given a symplectic manifold (M, ω) and a smooth function H : in terms of some chosen tensor field on E, such as a metric or symplec- M R, there is a unique smooth vector field, the Hamiltonian vector field tic form—but in general one can define G-structures that have nothing → directly to do with any tensor field. Indeed, the notion makes sense for XH Vec(M), defined by the condition ∈ any subgroup G GL(m, F). One should of course be aware that given G ⊂ ω(X , ) = dH. and a bundle E, it cannot be assumed that G-structures always exist—this H · − 2.5. INFINITE DIMENSIONAL BUNDLES 49 50 CHAPTER 2. BUNDLES isn’t even true for most of the examples above, e.g. a volume form exists Similarly, on any smooth vector bundle π : E M over a compact base → only if E is orientable. M, the spaces Ck(E) of k-times differentiable sections are Banach spaces.6 We can generalize one step further by considering not just subgroups of On the other hand, spaces consisting of smooth maps, e.g. C ∞(M, Rn) GL(m, F) but arbitrary groups that admit m-dimensional representations. and Γ(E), generally are not. This presents a bit of a technical annoyance, By definition, an m-dimensional (real or complex) representation of G is a because one would generally prefer to work only with spaces of smooth group homomorphism maps. We’ll use a standard trick (Exercise 2.64) to get around this problem ρ : G GL(m, F), in our example below. → thus associating with each element g G an m-by-m matrix ρ(g). The One thing to beware of in Banach spaces is that not all linear maps representation is called faithful if ρ is an∈injective homomorphism, in which are continuous. For two Banach spaces X and Y, we denote the space of F continuous linear maps X Y by (X, Y), with (X) := (X, X). An case it identifies G with a subgroup of GL(m, ). More generally, we → L L L can now make sense of the term G-valued transition map whenever an invertible map A : X Y is now called an isomorphism if both A and −1 → m-dimensional representation of G is given, since g G then acts on vec- A are continuous. The space (X, Y) admits a natural norm, defined ∈ L tors v Fm via the matrix ρ(g). The representation need not be faithful, by ∈ F Ax Y and G need not be a subgroup of GL(m, ). A L(X,Y) = sup k k , k k X x X x∈ \{0} k k and (X, Y) is then also a Banach space. 2.5 Infinite dimensional vector bundles WLe now summarize some facts about calculus in Banach spaces. This is With a little care, the theory of smooth manifolds and vector bundles can not the place to explore the subject in any detail, but the interested reader may consult the book by Lang [Lan93] to fill in the gaps. To start with, be extended to infinite dimensional settings, with quite powerful conse- the following definition should look very familiar. quences in the study of differential equations. We won’t go very deeply into this subject because the analysis is rather involved—but we can at Definition 2.57. Let X and Y be real Banach spaces, with U X an ⊂ least present the main ideas, and an example that should be sufficient open subset. Then a function F : U Y is differentiable at x U if motivation for further study. there exists a continuous linear map dF→(x) (X, Y) such that∈for all ∞ n ∈ L Infinite dimensional spaces are typically function spaces, e.g. C (M, R ), sufficiently small h X, the space of all smooth maps from a smooth compact manifold M to Rn. ∈ One of the most remarkable facts about differential calculus is that most of F(x + h) = F(x) + dF(x)h + h X η(h), k k · it can be extended in an elegant way to infinite dimensional vector spaces— where η is a map from a neighborhood of 0 X into Y with lim η(h) = however, the spaces for which this works are not arbitrary. We are most ∈ h→0 interested in the following special class of objects. 0. The linear operator dF(x) is called the derivative (or also linearization) of F at x. Definition 2.56. A Banach space is a vector space X with a norm The map F : U Y is continuously differentiable on U if dF(x) exists k k → such that every Cauchy sequence converges. for all x U and defines a continuous function dF : U (X, Y). ∈ → L Recall that a Cauchy sequence x X is any sequence for which the k Note that F is automatically continuous at x U if it is differentiable distances x x approach zero as∈both k and j go (independently) k j there. ∈ to infinity.k The− nonk trivial aspect of a Banach space is that given any The definition can be applied to complex Banach spaces as well by such sequence, there exists an element x X such that x x 0, k treating them as real Banach spaces. The derivative is then required to be i.e. x x. Metric spaces with this propert∈y are called completek − . k → k → only a real linear map.7 The requirement seems like nothing at first glance because, in fact, all finite dimensional normed vector spaces (real or complex) are Banach 6Technically, one should not call these Banach spaces but rather Banachable, because spaces. The spaces Ck(M, Rn) with k < are also Banach spaces when although they do become Banach spaces when given suitable norms, these norms are ∞ not canonical. One must then show that all suitable choices of norms are equivalent, given suitable norms, e.g. the standard norm on C1([0, 1], Rn) is meaning that they all define the same notion of convergence. 7 f = sup f(x) + sup f 0(x) . A map with a complex linear derivative is called complex analytic—the study of k k x∈[0,1] | | x∈[0,1] | | these is a different subject altogether. 2.5. INFINITE DIMENSIONAL BUNDLES 51 52 CHAPTER 2. BUNDLES

The derivative of a map from an open set into a Banach space has now Here Yα is a Banach space, and the restriction of Φα to each fiber is a been defined as another map from an open set into a Banach space. Thus Banach space isomorphism. Furthermore, these trivializations must be −1 the process can be iterated, defining higher derivatives and the notion of smoothly compatible, in that for α = β, Φβ Φα takes the form smooth maps U Y. Standard features of differential calculus such as 6 ◦ → (x, v) (x, g (x)v) the sum rule, product rule and chain rule all have elegant generalizations 7→ βα to this setting. wherever it is defined, and the transition maps The definitions so far make sense for all normed vector spaces, not just Banach spaces. On the other hand, the completeness condition plays gβα : U α U β (Yα, Yβ) a crucial role in proving the following generalization of a standard result ∩ → L from advanced calculus. are all smooth.

Theorem 2.58 (Inverse Function Theorem). Let X and Y be Banach This last point about the transition maps gβα contains some very serious spaces, U X an open subset, and F : U Y a smooth map such that analysis in the infinite dimensional case, for it is a stronger condition than for some x⊂ U with F(x) = y Y, the→derivative dF(x) : X Y just requiring that ∈ ∈ → is an isomorphism. Then there are open neighborhoods x U and ∈ U ⊂ U U y Y such that the restriction of F to is an invertible map . ( α β) Yα Yβ : (x, v) gβα(x)v ∈ V ⊂ U U → V ∩ × → 7→ Its inverse F−1 is also smooth, and be a smooth map. Unlike in finite dimensional calculus, where maps are usually presumed smooth until proven nonsmooth, it typically takes con- d(F−1)(y) = (dF(x))−1 : Y X. → siderable effort to prove that a map between Banach spaces is smooth. Thus the construction of a smooth Banach manifold or bundle structure There is similarly an infinite dimensional version of the implicit func- on any given object can be a rather involved process, even if one can see tion theorem, which can be treated as a corollary of the inverse function intuitively that certain spaces should be Banach manifolds. A very general theorem. We refer to [Lan93] for a precise statement, and the proofs of procedure for these constructions on so-called “manifolds of maps” was both results. presented in a paper by H. Eliasson [El˘ı67], whose ideas can be applied to We can now define smooth manifolds and vector bundles modeled on the example below. Banach spaces. Here we will present only a brief discussion of the main definitions—the recommended reference for this material is Lang’s geome- Example 2.61 (Stability of periodic orbits). Consider a smooth n- try book [Lan99]. dimensional manifold M with a smooth time-dependent vector field Xt which is 1-periodic in time, i.e. Xt is a smooth family of smooth vector Definition 2.59. A smooth Banach manifold is a Hausdorff topological fields such that X X . We wish to consider the orbits x : R M of U t+1 ≡ t → space M with an open covering M = α α and a system of charts the dynamical system x˙(t) = X (x(t)), (2.5) U S t ϕα : α Ωα, → and particularly those that are 1-periodic, satisfying x(t + 1) = x(t). Such a solution can be regarded as a map x : S1 M, where the circle S1 is where each Ωα is an open subset of some Banach space Xα, and the tran- t → −1 defined as R/Z. Denote by Φ : M M the flow of this system from time sition maps ϕβ ϕα are smooth wherever they are defined. → ◦ 0 to time t, i.e. Φt is a smooth family of diffeomorphisms such that for any t 8 Definition 2.60. M x0 M, x(t) := Φ (x0) is the unique orbit of (2.5) with x(0) = x0. Given a smooth Banach manifold , a Banach space ∈ bundle π : E M over M is a smooth Banach manifold E with a smooth Given any 1-periodic solution x(t) with x(0) = x0, we call the orbit → −1 surjective map π, such that each fiber Ex = π (x) for x M is a Banach nondegenerate if the linear isomorphism space, and there is a system of local trivializations (U ,∈Φ ) , where the { α α } 1 open sets U cover M, and Φ is a set of homeomorphisms dΦ (x0) : Tx0 M Tx0 M { α} { α} → 8Φt may not be globally well defined if M is noncompact, but this will cause no Φ : π−1(U ) U Y . problem in our discussion. α α → α × α 2.5. INFINITE DIMENSIONAL BUNDLES 53 54 CHAPTER 2. BUNDLES does not have 1 as an eigenvalue. This implies that the orbit is isolated, Thus T is a smooth Banach space bundle. It is contained in another B → B i.e. there is no other 1-periodic orbit in some neighborhood of this one (a smooth Banach space bundle , with fibers proof is outlined in Exercise 2.63). E → B = C0(x∗T M). Nondegeneracy has an even stronger consequence: that the orbit is Ex “stable” under small perturbations of the system. To state this precisely, All of these statements can be proved rigorously using the formalism of add an extra parameter to the vector field X , producing a smooth 2- t Eliasson [El˘ı67]. We now construct a smooth section s : as follows. parameter family of vector fields, 1-periodic in t, with the real number  For a C1-loop x : S1 M, s(x) will be the C0-vector fieldB →alongE x given varying over some neighborhood of 0 such that → by X0 X . s(x)(t) = x˙(t) Xt(x(t)). t ≡ t − The fact that this section is smooth also follows from [El˘ı67]. Clearly, This will be called a smooth 1-parameter perturbation of the time dependent the C1-solutions of the system (2.5) are precisely the zeros of the section vector field X . We aim to prove the following: t s : . B → E Theorem 2.62. Given any nondegenerate 1-periodic orbit x(t) of the sys- Exercise 2.64. k  Show that any C -solution to (2.5) is actually of class tem (2.5) and a smooth 1-parameter perturbation Xt of Xt, there is a Ck+1. By induction then, all C1-solutions are smooth! (This is the simplest unique smooth 1-parameter family of loops example of an elliptic bootstrapping argument.) x : S1 M We can now attack Theorem 2.62 via the infinite dimensional implicit →    function theorem. Let x be an orbit of (2.5), hence s(x) = 0. By for  in a sufficiently small neighborhood of 0, such that x˙ = Xt (x ) and ∈ B 0 choosing appropriate charts and local trivializations, we can identify the x (t) x(t). section s in a neighborhood of x with some smooth map ≡ ∈ B Exercise 2.63. The following argument proves both Theorem 2.62 and F : U Y, the previous statement that a nondegenerate periodic orbit is isolated. Let → ∆ M M denote the diagonal submanifold ∆ = (x, x) x M , and ⊂ × { | ∈ } where X and Y are Banach spaces, U X is an open neighborhood of 0 consider the embedding and F(0) = 0. We can also assume without⊂ loss of generality that the zero s−1 F−1 F : M M M : x (x, Φ1(x)). set (0) in a neighborhood of x corresponds to the set (0). → × 7→ Lemma 2.65. The linearization dF(0) : X Y is an isomorphism. Show that F intersects ∆ transversely at F (x ) ∆ if and only if the → 0 ∈ orbit x(t) is nondegenerate. The result follows from this via the finite We postpone the proof of this until Chapter 3, where we will develop dimensional implicit function theorem—fill in the gaps. (See [Hir94] for a convenient means of computing dF(0) via covariant derivatives—this is more on transversality and its implications.) where the assumption that M is orientable will be used. To see the impli- cations for stability of periodic orbits, consider now a smooth 1-parameter Alternative proof of Theorem 2.62. Leaving aside Exercise 2.63 for the mo-  perturbation Xt of the vector field Xt. This can be viewed as a 1-parameter ment, the result can also be proved by setting up the system (2.5) as a perturbation s of the section s, or equivalently, a smooth map smooth section of a Banach space bundle. Assume for simplicity that M is orientable (this restriction can be removed with a little cleverness). We F : ( δ, δ) U Y : (, u) F(u) − × → 7→ first ask the reader to believe that the space of maps where ( δ, δ) R is a sufficiently small interval and F0 F. The lin- − ⊂ e ≡ := C1(S1, M) earization dF(0, 0) : R X Y takes the form B ⊕ → admits a natural smooth Banach manifold structure, such that for each x e ∂F C1(S1, M), the tangent space T is canonically isomorphic to a Banac∈h dF(0, 0)(h, v) = h (0, 0) + dF(0)v, xB ∂ space of sections on the pullback bundle (cf. Example 2.35), e and it follows easilye from Lemma 2.65 that this operator is surjective, T = C1(x∗T M). with a one-dimensional kernel. Thus by the implicit function theorem, xB 2.5. INFINITE DIMENSIONAL BUNDLES 55 56 CHAPTER 2. BUNDLES some neighborhood of 0 in the set F−1(0) is a smooth one-dimensional 2.6 General fiber bundles submanifold of ( δ, δ) U. It follows that the same is true for the set − × e By now it should be apparent that vector bundles are worthy of study. (, y) ( δ, δ) s(y) = 0 , { ∈ − × B | } We next examine some objects that clearly deserve to be called “bundles,” and modulo a few analytical details, this implies Theorem 2.62. though their fibers are not vector spaces. The preceding may seem like an unwarranted amount of effort given Example 2.69 (Frame bundles). Recall that if π : E M is a smooth → that we already saw a more elementary proof in Exercise 2.63. The real vector bundle of rank m, a frame over some subset M is an ordered U ⊂ reason to consider this approach is that a similar argument can be applied set (v1, . . . , vm) of smooth sections vj Γ(E U ) which span each fiber Ex ∈ | to the study of certain systems of partial differential equations, where no for x . Restricting to a single fiber, a frame on Ex is simply an ordered ∈ U such finite dimensional method is available. We now look briefly at an basis of Ex. Denote by F Ex the set of all frames on Ex, and let example that is important in symplectic topology. F E = F Ex. Example 2.66 (Pseudoholomorphic curves). Given a manifold W of x∈M dimension 2n, an almost complex structure on W is a complex structure [ J Γ(End(T W )) on its tangent bundle, i.e. for each x W , J defines a This space has a natural topology and smooth manifold structure such that ∈ 2 ∈ any frame over an open subset M defines a smooth map F E. linear map TxW TxW with J = Id. This gives the tangent bundle U ⊂ U → T W W the structure→ of a complex− vector bundle with rank n (cf. We can also define a smooth projection map πF : F E M such that → −1 → −1 Section 2.4), and the pair (W, J) is called an almost complex manifold. For πF (x) = F Ex for all x M. It seems natural to call the sets πF (x) fibers, and π : F E M ∈is called the frame bundle of E. The fibers F E n = 1, an almost complex manifold (Σ, j) is also called a Riemann surface F → x Given a Riemann surface (Σ, j) and an almost complex manifold (W, J) are not vector spaces, but rather smooth manifolds. of dimension 2n, a map u : Σ W is called a pseudoholomorphic (or → Exercise 2.70. Show that for each x M, F Ex can be identified with J-holomorphic) curve if it satisfies the general linear group GL(m, F), though∈ not in a canonical way. T u j = J T u. (2.6) ◦ ◦ A section of the bundle F E M over some subset M is defined— predictably—as a map s : →F E such that π s = IdU ⊂. Thus a smooth Working in local coordinates on both manifolds, (2.6) becomes a sys- U → ◦ U tem of 2n first order partial differential equations in two variables (since section s : F E is precisely a frame over for the vector bundle E, and we denoteU →the space of such sections by Γ(UF E ). Two things must dim Σ = 2), called the nonlinear Cauchy-Riemann equations. The next |U exercise explains the reason for this terminology. be observed:

Exercise 2.67. If (Σ, j) = (W, J) = (C, i), show that a map f : C C (i) Γ(F E U ) is not a vector space, though it does have a natural topology, | satisfies (2.6) if and only if it is an analytic function. → i.e. one can define what it means for a sequence of smooth sections s Γ(F E ) to converge (uniformly with all derivatives) to another The infinite dimensional implicit function theorem can be applied to k ∈ |U smooth section s Γ(F E ). the study of J-holomorphic curves just as in Example 2.61. ∈ |U (ii) Sections s : F E can always be defined over sufficiently small Exercise 2.68. Without worrying about the analytical details, write down U → 1 open subsets M, but there might not be any global sections a Banach space bundle over the Banach manifold := C (Σ, W ) and U ⊂ a section whose Ezeros are precisely the J-holomorphicB curves u : s : M F E. Indeed, such a section exists if and only if the vector B → E → (Σ, j) (W, J) of class C1. (As in Exercise 2.64, it turns out that any such bundle E M is trivializable. → → solution is actually a smooth map, due to elliptic regularity theory. Note Example 2.71 (Unit sphere bundles and disk bundles). Given a real that technically, Σ must be compact in order to apply Eliasson’s formalism, vector bundle π : E M of rank m with a smooth bundle metric , , → h i but there are various ways to work around this for certain noncompact define domains.) Sm−1E = v E v, v = 1 . { ∈ | h i } Much more on J-holomorphic curves can be found in the book by Mc- Then the restriction of π to Sm−1E is another surjective map p : Sm−1E → Duff and Salamon [MS04]. M, whose fibers p−1(x) are each diffeomorphic to the sphere Sm−1. 2.6. GENERAL FIBER BUNDLES 57 58 CHAPTER 2. BUNDLES

A section s : M Sm−1E is simply a section of the vector bundle Notice that the coordinate vector field V (t, x) = ∂ on R N descends → ∂t × E whose values are unit vectors. Unit sphere bundles can sometimes be to a well defined smooth vector field V on the total space Mϕ, whose 1- useful as alternative descriptions of vector bundles: they carry all the same periodic orbits can be identified with the fixede points of ϕ. In this way the information, but have the nice property that if M is compact, so is the total mapping torus can be used to analyze fixed points by changing a discrete space Sm−1E. In particular, if (M, g) is a compact Riemannian n-manifold, dynamical system into a continuous dynamical system. then S1T M is a compact (2n 1)-manifold, and the geodesic equation on − This next example similarly has no direct relation to any vector bundle. M defines a smooth vector field on S1T M. A closely related object is the disk bundle Example 2.73 (The Hopf fibration). Regard S3 as the unit sphere in C2, and define a map with values in the extended complex plane by m D E = v E v, v 1 , z { ∈ | h i ≤ } π : S3 C : (z , z ) 1 . 1 2 z with fibers that are smooth compact manifolds with boundary. Such ob- → ∪ {∞} 7→ 2 jects arise naturally in the construction of tubular neighborhoods: given a Using the stereographic projection to identify C with S2, the map ∪ {∞} manifold M and a closed submanifold N M, it turns out that a neigh- π : S3 S2 turns out to be smooth, and its fibers π−1(z) are sets of the ⊂ → borhood of N in M can always be identified with a disk bundle over N, form eiθ(z , z ) θ R for fixed (z , z ) S3 C2. Thus we have a { 1 2 | ∈ } 1 2 ∈ ⊂ with fibers of dimension dim M dim N (cf. [Bre93]). fiber bundle π : S3 S2 with fibers diffeomorphic to S1. Such objects − are called circle bund→les, and this one in particular is known as the Hopf Clearly it is useful to consider bundles whose fibers are all diffeomorphic fibration. It plays an important role in homotopy theory—in particular, π manifolds, varying over the base in some smooth way. Each of the examples is the simplest (and in some sense the archetypal) example of a map from so far has been defined in terms of a given vector bundle, but this needn’t S3 to S2 that cannot be deformed continuously to the identity. The proof be the case in general, as the next example shows. of this fact uses a general result that relates the higher homotopy groups Example 2.72 (Mapping tori). Suppose N is a smooth n-manifold and of the total space, fiber and base of any fiber bundle. For a topologist, this ϕ : N N is a diffeomorphism. Then we define the mapping torus of ϕ gives more than sufficient motivation for the study of fiber bundles. → to be the smooth (n + 1)-manifold Here now are the definitions—we once again begin in a purely topolog- ical context, and subsequently incorporate the notion of smoothness. Mϕ = (R N)/ , × ∼ Definition 2.74. A fiber bundle consists of three topological spaces E, M with the equivalence relation defined by (t, x) (t + 1, ϕ(x)). The projec- and F , called the total space, base and standard fiber respectively, together tion R N R then descends to a smooth surjectiv∼ e map with a continuous surjective map π : E M such that for every point x × → M there exists an open neighborhood x→ M and a local trivialization∈ 1 ∈ U ⊂ π : Mϕ S := R/Z, → Φ : π−1( ) F. U → U × with the “fibers” (M ) := π−1([t]) all diffeomorphic to N. To see that ϕ [t] Here Φ is a homeomorphism that takes π−1(x) to x F for each x . the fibers vary “smoothly” with respect to t S1, we can construct “local { } × ∈ U −1 ∈ 1 −1 trivializations” Φ : π ( ) N over open subsets S . For The fibers Ex := π (x) of a fiber bundle are thus homeomorphic to U → U × 1 U ⊂ example, regarding 1 := (0, 1) as an open subset of S , there is the obvious some fixed space F , by homeomorphisms that vary continuously over the diffeomorphism Φ :Uπ−1( ) N defined by base. 1 U1 → U1 × Remark 2.75. Geometers often use the terms fibration and fiber bundle Φ ([(t, x)]) = (t, x) for t (0, 1). 1 ∈ synonymously, though the former has a slightly more general meaning in 1 3 topology, cf. [Hat02]. The same expression with t 2 , 2 gives a well defined diffeomorphism Φ : π−1( ) N for the∈ open set := 1 , 3 S1. Observe that Definition 2.76. Suppose π : E M is a fiber bundle with standard fiber 2 U2 → U2 ×
U2 2 2 ⊂ → covers S1, thus the two trivializations , Φ determine F , such that M and F are both smooth manifolds. A smooth structure on 1 2 j
j j∈{1,2} whatU ∪ Uwe will momentarily define as a “smooth fib{Uer bundle} structure” for π : E M is a maximal set of local trivializations , Φ such that the → {Uα α} π : M S1. open sets cover M and the maps Φ are diffeomorphisms. ϕ → {Uα} { α} 2.7. STRUCTURE GROUPS 59 60 CHAPTER 2. BUNDLES

We see from this definition that a smooth fiber bundle has each fiber dif- Most of these examples are infinite dimensional,9 which introduces some feomorphic to the standard fiber F , by diffeomorphisms that vary smoothly complications from an analytical point of view. They cannot be considered over M. It’s not hard to show from this that the total space E is then a smooth Lie groups, though they clearly are topological groups. smooth manifold of dimension dim M + dim F . Given a topological group G and a topological space F , we say that G The terms section and fiber preserving map have the same definition in acts continuously from the left on F if there is a continuous map this context as for a vector bundle. G F F : (a, x) ax × → 7→ 2.7 Structure groups satisfying ex = x and (ab)x = a(bx) The version of a fiber bundle defined above is somewhat weaker than the for all a, b G and x F , where e G is the identity element. The definition given in some texts, notably the classic book by Steenrod [Ste51]. ∈ ∈ ∈ The stricter definition includes an extra piece of structure, called the struc- simplest example is the natural action of any subgroup G Homeo(F ) on F , defined by (ϕ, x) ϕ(x). ⊂ ture group of a bundle. To explain this, we will need to assume some 7→ knowledge of topological groups and Lie groups, which are discussed in Similarly, a right action Appendix B. F G F : (x, a) xa The most important Lie groups are of course the linear groups: GL(n, F), × → 7→ SL(n, F), O(n), SU(n), Sp(n) and so forth, all of which arise naturally in must satisfy the study of vector bundles with extra structure such as bundle metrics or xe = x and x(ab) = (xa)b. volume forms. In the study of general fiber bundles, one also encounters various “infinite dimensional groups,” such as the following examples. For For example, a subgroup G Homeo(F ) defines a right action on F by ⊂ any topological space F , there is the group (x, ϕ) ϕ−1(x). For any left or right action, the operation of any single group 7→element a G on F defines a homeomorphism. Homeo(F ) = ϕ : F F ϕ is bijective, ϕ and ϕ−1 both continuous . ∈ { → | } You should take a moment to convince yourself that the definitions of “left action” and “right action” are not fully equivalent: the key is the On an oriented topological manifold, this has an important subgroup associativity condition. For instance, the map (x, ϕ) ϕ−1(x) mentioned 7→ Homeo+(F ) = ϕ Homeo(F ) ϕ preserves orientation ; above defines only a right action, not a left action. { ∈ | } On the other hand, any left action (a, x) ax defines a corresponding 7→ we refer to [Hat02] for the definition of “orientation preserving” on a topo- right action by (x, a) a−1x, and vice versa. In this way one can switch 7→ logical manifold. If F is a smooth manifold, we have the corresponding back and forth between left and right actions for the sake of notational subgroups convenience; the distinction is essentially a matter of bookkeeping. Remark 2.77. The following example shows why it can be convenient to Diff(F ) = ϕ Homeo(F ) ϕ and ϕ−1 are smooth , { ∈ | } consider both left and right actions in the same discussion. For some and in the oriented case, topological group G, let F = G and consider the natural actions G F F and F G F defined by group multiplication. These define× + → × → Diff (F ) = ϕ Diff(F ) ϕ preserves orientation . homeomorphisms of F by La(h) = ah and Rb(h) = hb for every a, b G, { ∈ | } ∈ and due to the associative law, La Rb = Rb La. This will be useful when There are still other groups corresponding to extra structures on a smooth we define the group action on frame◦ bundles◦in Section 2.8. manifold F , such as a Riemannian metric g or a symplectic form ω. These The terms smooth left/right action now have obvious definitions if G is two in particular give rise to the groups of isometries and symplectomor- a Lie group and F is a smooth manifold. In this case, each element a G phisms respectively, defines a diffeomorphism on F . ∈ ∗ Isom(F, g) = ϕ Diff(F ) ϕ g g , 9 { ∈ | ≡ } The exception is Isom(F, g), which is generated infinitessimally by the finite dimen- Symp(F, ω) = ϕ Diff(F ) ϕ∗ω ω . sional space of Killing vector fields; see e.g. [GHL04]. { ∈ | ≡ } 2.7. STRUCTURE GROUPS 61 62 CHAPTER 2. BUNDLES

We are now ready to relate all of this to fiber bundles. For a gen- Example 2.80 (Vector bundles with structure). A smooth vector eral bundle π : E M, it’s clear that any two local trivializations bundle π : E M of rank m can now be defined as a smooth fiber bundle −1 → −1 → m m Φα : π ( α) α F and Φβ : π ( β) β F can be related with standard fiber F and structure group GL(m, F), which acts on F by U → U × U → U × in the natural way. The notion of a GL(m, F)-bundle isomorphism is then −1 Φβ Φα (x, p) = (x, gβα(x)p) (2.7) equivalent to our previous definition of vector bundle isomorphism. ◦ The group reduces further if the bundle is given extra structure: for in- where g : Homeo(F ) is a continuous “transition map,” defined βα α β stance, a Euclidean or Hermitian bundle metric reduces GL(m, F) to O(m) in terms of Uthe∩naturalU → left action of Homeo(F ) on F . At the most basic or U(m) respectively, while a volume form yields SL(m, F). The combina- level then, we can say that the bundle π : E M “has structure group tion of a Hermitian metric and compatible complex volume form reduces Homeo(F )”. → the structure to SU(m). There are corresponding statements about sym- Definition 2.78. Suppose π : E M is a fiber bundle, with standard plectic structures (Sp(m)), indefinite metrics (O(m k, k)) and so forth. − fiber F , and G is a topological group.→ A reduction of the structure group Indeed, a vector bundle with a G-structure for any Lie group G can be Fm of this bundle to G is a system of local trivializations α, Φα such that defined as a fiber bundle with standard fiber and structure group G, {U } Fm α is an open covering of M, and the maps Φα are related to each other where G acts on via some given group representation. Each such struc- as{Uin} (2.7) by a family of continuous transition maps ture comes with its own definition of a bundle isomorphism; for instance, a smooth vector bundle isomorphism on a Euclidean bundle is an O(m)- gβα : α β G, bundle isomorphism if and only if its restriction to each pair of fibers is an U ∩ U → isometry. acting on the trivializations via some continuous left action G F F . When π : E M is a smooth fiber bundle and G is a Lie group× →acting Example 2.81 (Frame bundles with structure). In addition to reduc- smoothly on the→standard fiber F , we additionally require the transition ing the structure group to a subgroup of GL(m, F), the aforementioned maps to be smooth. structures on vector bundles each define special frame bundles that have If such a reduction exists, we say that the bundle π : E M has the same structure group. For example, given a Euclidean vector bundle → (E, , ) over M, we can define the orthogonal frame bundle F O(m)E M, structure group G. A fiber bundle with structure group G is called a G- h i → bundle. a subbundle of F E which contains only orthonormal frames. The standard fiber of F O(m)E is the Lie group O(m), which admits a natural action of For a fiber bundle with extra structure, it is appropriate to introduce O(m), defined by the group multiplication. Clearly any two trivializations a somewhat restricted notion of bundle isomorphism. can then be related by a transition map with values in O(m). The same remarks apply to any reduction of the structure group of a Definition 2.79. Suppose π : E M and π : E M are two 1 1 2 2 vector bundle: there is always a corresponding frame bundle, consisting of G-bundles with the same standard fib→er F and the same G→-action on F . the frames that are “preferred” by the structure in question, e.g. symplectic Then a G-bundle isomorphism is a fiber preserving homeomorphism (or frames, unitary frames, frames of unit volume, etc.. The frame bundle diffeomorphism) f : E E such that for every pair of local trivializations 1 2 always has the same structure group as the vector bundle, and its fibers Φ : π−1( ) F→and Ψ : π−1( ) F , there is a continuous α 1 Uα → Uα × β 2 Uβ → Uβ × are also diffeomorphic to this group. We’ll have more to say about this in (or smooth) map f : G with βα Uα ∩ Uβ → the next section, on principal bundles. −1 m Ψβ f Φα (x, p) = (x, fβα(x)p). Example 2.82 (Unit sphere bundles). For the bundle S −1E M de- ◦ ◦ → fined in Example 2.71 with respect to a Euclidean vector bundle (E, , ), Steenrod [Ste51] regards the structure group as part of the intrinsic there is a natural reduction of the structure group to O(m). Namelyh , wi e structure of a fiber bundle, included in the definition. In principle, our restrict to orthonormal frames on E and use these to define local trivial- definition is equivalent to his simply by naming Homeo(F ) as the structure izations of Sm−1E, so that transition maps in O(m) act on the unit sphere group—but this is beside the point, because many of the most interesting Sm−1 Rm. cases are those where the structure group can be reduced to a simpler ⊂ group G, particularly if G is a finite dimensional Lie group. Example 2.83 (Symplectic fibrations). We now give an example where It’s time for some examples. the structure group is an infinite dimensional subgroup of Homeo(F ). A 2.8. PRINCIPAL BUNDLES 63 64 CHAPTER 2. BUNDLES symplectic fibration is a smooth fiber bundle π : E M whose standard with structure group G. These bundles have the special property that → fiber is a symplectic manifold (F, ω), with structure group Symp(F, ω). their standard fiber and structure group are the same Lie group G, which This last condition gives each fiber Ex the structure of a symplectic man- acts from the left on itself by group multiplication. There is also a natural ifold, such that the local trivializations restrict to symplectomorphisms fiber preserving action of G on F GE: to see this, consider the example E x F . Thus E can be thought of as a bundle of symplectic man- of the orthonormal frame bundle F O(m)E for a real vector bundle E with x → { } × ifolds that are all related by smoothly varying symplectomorphisms. A Euclidean metric , . Given any orthonormal frame p = (v1, . . . , vm) 1 2 O(m) h i i ∈ fiber preserving diffeomorphism f : E E between two such bundles is (F E)x for Ex and a matrix A O(m) with entries A j, another such → 1 2 O(m) ∈ a Symp(F, ω)-bundle isomorphism if and only if its restriction to Ex Ex frame p A (F E)x is obtained by right multiplication, for each x M is a symplectomorphism. → · ∈ ∈ 1 1 For a very simple example, one can take any symplectic manifold (F, ω) A 1 A m . ·.· · . i i with a symplectomorphism ϕ Symp(F, ω) and define the mapping torus p A = v1 vm . .. . = (viA 1, . . . , viA m). ∈ 1 · · · ·   Mϕ as in Example 2.72. This is now a symplectic fibration over S . Am Am
 1 · · · m   We can now give the simplest possible definition of “orientation” for a Similar remarks apply to bundles of unitary frames, oriented frames and fiber bundle. To simplify matters, we restrict to the smooth case. so forth. Since frame bundles play a helpful role in understanding vector bundles Definition 2.84. For a smooth fiber bundle π : E M with standard with extra structure, we give a special name to the class of fiber bundles fiber F , an orientation of the bundle is a reduction of→its structure group that share the same special properties. The following is the first of two from Diff(F ) to Diff+(F ). equivalent definitions we will give—it applies to smooth bundles, though Take a moment to convince yourself that this matches the old definition it should be clear how to generalize it to topological bundles and groups. in the case of a real vector bundle. The next step is of course to give an Definition 2.86. A smooth principal fiber bundle π : E M is a smooth example of a non-orientable fiber bundle—it just so happens that the most fiber bundle whose standard fiber is a Lie group G, such that→ the structure popular example is also the best known non-orientable manifold in the group reduces to G, acting on itself by left multiplication. world. These are sometimes simply called principal bundles, or principal G- Example 2.85 (The M¨obius strip). Recall the non-orientable line bun- 1 1 bundles if we want to specify the structure group. dle ` S from Example 2.12, where S is identified with the unit circle The definition above is analogous to the notion of a vector bundle as C → R2 in and the fibers are lines in , specifically a fiber bundle with standard fiber Fm and structure group GL(m, F) (see Example 2.80). This is, of course, not the most intuitive definition possible, cos(θ/2) 2 `eiθ = R R . which is why we originally characterized vector bundles in terms of the sin(θ/2) ⊂   intrinsic linear structure on the fibers. An analogous definition is possible for principal bundles π : E M, as soon as we understand what intrinsic We can define a Euclidean structure on this bundle using the natural inner → product of R2, and then consider the corresponding “unit disk bundle” structure the fibers possess. For each x M, Ex is diffeomorphic to a Lie group G, but not in a canonical way∈ —in particular, there is no 1 D E = v ` v, v 1 , special element e Ex that we can call the identity. On the other hand, ∈ { ∈ | h i ≤ } having reduced the structure group to G, we’ve chosen a special set of local 1 −1 a smooth fiber bundle over S with standard fiber [ 1, 1]. The total space trivializations Φα : π ( α) α G that are related to each other on 1 − U → U × of D E is the M¨obius strip. each fiber by the left action of G. This has the effect of inducing a natural fiber preserving right action

2.8 Principal bundles E G E : (p, g) pg, × → 7→ We’ve already seen some examples of principal fiber bundles, namely the defined in terms of any local trivialization Φ (p) = (π(p), f (p)) α α ∈ Uα × frame bundles F GE M corresponding to any vector bundle E M G by f (pg) = f (p)g. This is independent of the choice because the → → α α 2.8. PRINCIPAL BUNDLES 65 66 CHAPTER 2. BUNDLES right action of G on itself commutes with the left action of the transition Proposition 2.88. A smooth principal bundle π : E M is trivial if and → functions (cf. Remark 2.77); indeed if p E for x , only if it admits a global smooth section s : M E. ∈ x ∈ Uα ∩ Uβ →

fβ(pg) = gβα(x)fα(pg) = gβα(x)fα(p)g = fβ(p)g. This is of course quite different from the situation in a vector bundle, which always admits an infinite dimensional vector space of global smooth Clearly the action restricts to a right action of G on each fiber Ex, and sections. For principal bundles there is a one-to-one correspondence be- one easily sees that this is identical to the group action on frames that we tween sections and trivializations. A consequence is that any smoothly defined above for the special case when E is a frame bundle. Two other varying group structure on the fibers would trivialize the bundle, since important properties are worth noting: there would then be a global section s : M E, taking every point to the → identity element in the corresponding fiber. (i) G acts freely, i.e. without fixed points, meaning pg = p for some p E and g G if and only if g is the identity. Equivalently, G acts Definition 2.89. Given two principal G-bundles E and E over the same ∈ ∈ 1 2 freely on each fiber Ex. base M, a principal bundle isomorphism is a smooth map Ψ : E1 E2 which is fiber preserving and G-equivariant, i.e. Ψ(pg) = Ψ(p)g. → (ii) G acts transitively on each fiber: for any p, q E , there exists g G ∈ x ∈ such that q = pg. Exercise 2.90. Notice that the definition above does not explicitly require Ψ to be an invertible map. Show however that every principal bundle Conversely, suppose we have a smooth fiber bundle π : E M and a isomorphism is automatically invertible, and its inverse is another principal Lie group G with a fiber preserving smooth right action E G→ E that × → bundle isomorphism. restricts to a free and transitive action on each fiber. These two properties of the action imply that each fiber is diffeomorphic to G. In fact, for any Exercise 2.91. Show that a smooth map between principal G-bundles is x M, we can choose an open neighborhood x M and a smooth ∈ ∈ U ⊂ a principal bundle isomorphism if and only if it is a G-bundle isomorphism section s : E, and use it to define the inverse of a local trivialization, U → (see Definition 2.79). Φ−1 : G π−1( ) : (x, g) s(x)g. Clearly, a principal bundle is trivializable if and only if it is isomorphic U × → U 7→ to the trivial principal bundle Now choose an open cover M = with smooth sections s : E, α Uα α Uα → and use these to define a system of local trivializations Φα, α in the π : M G M, same manner. For x thereS is a unique element g{ (xU) }G such × → ∈ Uα ∩ Uβ βα ∈ that sα(x) = sβ(x)gβα(x), and one could use an implicit function theorem where the right action is (x, h)g := (x, hg). argument to show that the map g : G is smooth. Then we As we’ve seen already, any vector bundle π : E M with structure βα α β → have U ∩ U → group G yields a principal G-bundle F GE M, the frame bundle. Prin- cipal bundles arise naturally in other contexts→ as well. −1 Φβ Φα (x, g) = Φβ(sα(x)g) = Φβ(sβ(x)gβα(x)g) = (x, gβα(x)g), ◦ Example 2.92 (The Hopf fibration). Recall the bundle π : S3 S2 so g are indeed the transition maps for the system , Φ . We’ve from Example 2.73, with standard fiber S1, where S3 is regarded as→the βα {Uα α} proved that the following is equivalent to Definition 2.86. unit sphere in C2. The right action

Definition 2.87. 3 3 iθ iθ A smooth principal fiber bundle is a smooth fiber bundle S U(1) S : ((z1, z2), e ) e (z1, z2) π : E M together with a Lie group G and a fiber preserving right action × → 7→ E G→ E which restricts to each fiber freely and transitively. gives the Hopf fibration the structure of a principal U(1)-bundle. × → Thus each fiber has the intrinsic structure of a smooth manifold with Example 2.93 (An SO(n)-bundle over Sn). Regarding Sn as the unit a free and transitive G-action; as a result, the fiber must be diffeomorphic sphere in Rn+1, the natural action of SO(n + 1) on Rn+1 restricts to a tran- n n to G. This is the closest we can come to giving the fibers an actual group sitive action on S . Let x0 = (1, 0, . . . , 0) S , and define the surjective structure without requiring the bundle to be trivial—indeed, by the same map ∈ trick we used to construct local trivializations above, we have: π : SO(n + 1) Sn : A Ax . → 7→ 0 2.8. PRINCIPAL BUNDLES 67 68 CHAPTER 2. BUNDLES

Using the injective homomorphism [Spi65] M. Spivak, Calculus on manifolds: A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. 1 0 SO(n) , SO(n + 1) : B , [Spi99] , A comprehensive introduction to differential geometry, 3rd ed., Vol. 1, → 7→ 0 B   Publish or Perish Inc., Houston, TX, 1999. we can regard SO(n) as a subgroup of SO(n + 1) and define the natural [Ste51] N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, N. J., 1951. right action

SO(n + 1) SO(n) SO(n + 1) : (A, B) AB. × → 7→ Now observe that SO(n) is precisely the subgroup of matrices in SO(n + 1) that fix the point x . Thus if A π−1(x) for any x Sn, we have Ax = x 0 ∈ ∈ 0 and ABx0 = Ax0 = x, so the right action is fiber preserving. It is now simple to verify that the action is free and transitive on each fiber, thus π : SO(n + 1) Sn with this action is a principal SO(n)-bundle. As with the Hopf fibration,→ this construction is useful for understanding the topology of the spaces involved, particularly the special orthogonal groups. Similar constructions can be made for the unitary and special unitary groups. Exercise 2.94. Construct a principal U(n)-bundle over S2n+1 with total space U(n + 1).

References

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