Appendix A Tangent Bundles and Vector Bundles Let M be a smooth manifold. Let TM = UpEMTpM. We then have a set theoretic map 7r from T M to M given by 7r( v) = p if v E TpM. We wish to endow T M with a smooth structure. We fix a smooth atlas for M. For (U, </», a local chart in the atlas, let D = 7r-l(U) = UpEuTpM. We define by setting ¢(v) = (Xl(7r(V)), ... ,xm(7r(v)), V(Xl), ... , v(xm)) E 1R2m. Then ¢ is one-one on 7r-l(U) onto </>(U) x IRm, an open subset of 1R2m. We define local coordinates ~j on D by for 1 :::; j :::; 2m where Uj stand for the usual coordinates on 1R2m. That is, ~j(v) = {(Xj(7r(V)) for 1 :::; j :::; m; V(Xi) for m < j = m + i :::; 2m. We now show that this collection {(D, ¢)}, as U varies over all charts of the atlas for M, forms a smooth atlas for T M. The only thing to be checked is that the overlaps are smooth. Notice that (D, ¢) and (V, 7;;) overlap if and only if the correspond­ ing charts (U, </» and (V,7/1) of the atlas overlap. Let Xi and Yi be the corresponding local coordinates on U and V respectively. Then given (a, b) E ;f(U n V) x IRm we see that ~-1 ~ a I 7/1 (a, b) = ~bia. .p-l(a)" ;=1 y, 282 A. Tangent Bundles and Vector Bundles T2 p!..ove the smoothness~of the overl~s it is enough to show that Ui 0 </> 0 t/J-l is smooth from t/J(U n V) to </>(U n V). Let 1 $ i $ m. Then Ui 0 ¢ 0 ;f-l(a, b) = Xi 0 7r 0 ;f-l(a, b) = Xi(t/J-l(a)). Thus (Ul 0 ¢ 0 ;f-l, ... , Urn 0 ¢ 0 ;f-l) = </> 0 t/J-l. Hence the functions Ui 0 ¢ 0 ;f-l for 1 $ i $ m are smooth. Now, for j = m + r with 1 $ r $ m, we have That is, these are the components of the Jacobian of the map ¢ 0 ;j;-l. Thus these maps are also smooth. o This smooth manifold T M is called the tangent bundle of the given manifold M. Exercise A.I Show how to construct the cotangent bundle T* M = UpEMT;M. Exercise A.2 The map 7r : T M ~ M is smooth. It is called the canonical map of the tangent bundle T M to the base manifold M. Exercise A.3 Let </> : M ~ N be a smooth map. Then we have the derivative map D</> : TM ~ TN given by D</>(v) := D</>(p)(v) for v E TpM. Is this map smooth? Definition A.I A smooth section q of T M is a smooth map q : M ~ TM such that q(p) E TpM for all p E M. Notice that this is same as requiring that 7r 0 q(p) = p for all p EM. We usually denote sections by symbols like X, Y, etc. We claim that a smooth section of T M on M is a smooth vector field on M. The only thing to be verified is the smoothness of a section X of T M when it is considered as a vector field. This follows from the 283 Exercise A.4 Show that in the notation as above X is smooth if and only iffor any local chart (U, x) if we write X = E:'l X(xd 8~i on U (using the basis theorem) then the coefficients X(Xi) are smooth on U. The tangent bundle is an example of the following more general con­ cept. Definition A.2 A vector bundle over a (smooth) manifold M is a pair (E,7r) satisfying the following conditions: (a) E is a smooth manifold, called the total space. (b) 11': E -+ M is a smooth-map, called the projection. (c ) There is a fixed r E N (called the rank of E) such that for each p E M, 7r-1(p) is an r-dimensional vector space over R 7r-1(p) is called the fiber over p and is usually denoted by Ep. (d) Condition of Local Triviality: For each p E M, there is a neigh­ borhood U of p and a diffeomorphism such that for any fixed q E U, the map r 1-+ cp(q, v) is a linear isomorphism of]Rr onto the fiber Eq • On any manifold M there are many vector bundles, for example, the tangent bundle T M, the cotangent bundle T* M and more generally Tk M, the k-th tensor bundle on M (to be defined later). There are also trivial bundles (M x ]Rr, 7r) where 7r is the projection of M x ]Rr onto the first factor. Thus the condition (d) of Definition A.2 says that any vector bundle is locally trivial. Defiilition A.3 A (smooth) section of (E,7r) (or simply E) over M is a (smooth) map s : M -+ E such that s(p) E Ep. Thus a (smooth) vector field is a smooth section of TM. A section of any trivial bundle M x ]Rr is nothing but a map 8 : M -+ ]Rr. Definition A.4 A local frame of E is by definition sections {ei} defined on an open subset U of M such that {ei(p) : 1 ~ i ~ r} is a basis for the vector space Ep for all p E U. 284 A. Tangent Bundles and Vector Bundles Notice that, by local triviality of E, local frames always exist. How­ ever there may not exist even a single nonzero section (that is, a section s : M -+ E such that s(p) f:. 0 for all p EM). Can you think of an example? There is a natural bijection between local frames and local trivializa­ tion as in condition (d). Let (U, ip) be given as in (d). We fix a basis Vi oflRr and define ei(p):= ip(p,Vi). Then it is trivial to see that {ei} is a local frame on U. Conversely, if {ei} is a local frame on V, then we have a local trivialization of E on V as follows: If e E E, we have the expression e = I:i aiei(p). We then define 1jJ-l : 1l"-l(V) -+ V x IRr by setting 1jJ-l(e):= (p,al, ... ,ar ). Thus a vector bundle is trivial if and only if there exists a global frame, that is, a frame whose domain of definition U = M. We say a manifold is parallelizable if the tangent bundle T M is trivial. Thus TG, the tangent bundle of a Lie group is parallelizable whereas there exist manifolds which are not parallelizable. (Substantiate these claims.) There are certain obvious algebraic constructions we can carry out on the class of vector bundles on M. Given vector bundles E and F we can construct E*, Tk E, E EEl F, E ~ F and Tr E ~ TS F where Tk E denotes the vector bundle whose fibers are thek-th tensor powers of the fibers of E, which were introduced in Chapter 3. The reader should be able to construct these objects quite easily. Another important concept in the study -of vector bundles is the notion of transition functions. They arise as follows. Given a vector bundle Eon M, by local triviality, there exists an open covering {U",} of M such that we have the trivializations: ip", : U", x IRr -+ 1l"-l(U",) C E satisfying condition (d). Now if U", n Uf3 f:. 0, then we have the diffeo­ morphisms: ip", : (U", n U(3) x IRr -+ 1l"-l(U", n U(3) ipf3 : (U", n U(3) x IRr -+ 1l"-l(U", n U(3). As remarked above each ip", gives rise to a local frame {ef}i=l on U",. Now for p E U", n Uf3, both {ei(p)}i=l and {e1(p)}i=1 form a basis of Ep. Hence there exists gf3", E GL(r, 1R) which takes the a-basis to the /3-basis. In view of the fact that ip", are diffeomorphisms, it is easily seen that the map p t-+ gf3",(p) from U", n Uf3 to GL(r,lR) is smooth. These gf3", are called the transitions or transition functions of E. They enjoy the following properties: 285 1. go.o. = 1, the identity. -1 2. go.{3 = g{3o.. 3. If Uo. n U{3 n U·Y =1= 0, then g,,(o. = g"({3 0 g{3o. on the open set Uo. n U{3 n U"(. Conversely, if we are given an open covering {U a.} of a manifold M and smooth maps g{3o. : Uo. n U{3 -+ GL(r, JR) for all Q and (3 with Ua. n U(3 =1= 0 satisfying the above three properties, then we can construct a vector bundle E in a natural way. (Exercise.) Thus the transition functions gives us an indication as to how the locally trivial products Uo. x JRT are glued together. Appendix B Partitions of Unity Definition B.l A collection {Ua } of subsets of a topological space M is said to be locally finite if for all p E M there exists a neighbourhood U of p such that Un Ua = 0 except for finitely many indices a. Definition B.2 A partition of unity on a manifold M is a collection {gj : i E I} of smooth functions such that 1.