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, (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, 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. the collection of supports {supp gi : i E I} is locally finite.
2. gi ~ 0 on M for all i E I.
3. For all p EM, I:i gi(p) = 1. Note that by (1) this sum is finite.
A partition of unity {gd is said to be subordinate to an open cover {Ua } if for each i E I there exists an ai such that supp gi ~ Uai • We say that it is subordinate to the cover {Ui : i E I} with the same index set as the partition of unity if supp gi ~ Ui for all i E I.
Definition B.3 Let M be a topological space. We say that a cover {Ua } is a refinement of another cover {Vi} if for each a there is an i such that Ua ~ Vi.
Definition B.4 A topological space is paracompact if every open cover has a refinement consisting of open sets and the latter cover is locally finite.
Lemma B.l Any second countable, locally compact Hausdorff space M is paracompact. 287
Proof We first prove that there is a countable collection {Gi : i E N} of open sets in M with the following property:
Gi is compact, G i ~ GHl, M = U Gi • iEN Let {Ud be a countable basis of M consisting of open sets with compact closures. Such a basis can be obtained from starting with any countable basis and selecting the sub collection consisting of basic open sets with compact closures. The fact that M is Hausdorff and second countable shows that this collection is a basis. Now let G1 = U1• Assume that Gk = U1 u· .. U Ui/o. Let jk+l be the smallest integer greater than jk such that
G k c_ Uik+lUi=1 i·
Define Gk+1 = u1~ilUi. Then {Gd satisfies Equation (*) above. Now let {Uo : a E A} be an arbitrary cover. The set Gi \ Gi - 1 is compact and Gi \ Gi - 1 ~ GH1 \ Gi - 2 , an open set. For each i ~ 3, we choose a finite sub cover of the open cover {Uo n GH1 \ Gi - 2 : a E A} of Gi \ Gi-l. We also choose a finite sub cover of the open cover {Uo n Ga : a E A} of the compact set G2 • This collection of open sets is easily seen to be a countable locally finite refinement of {Uo }, and consists of open sets with compact closure. D
Theorem B.2 Any second countable smooth manifold admits a parti• tion of unity subordinate to a given smooth atlas of M.
Proof Let {Uo } be an open cover of M. Then there exists a locally finite refinement consisting of coordinate charts (Vi, Notice that the sum in the denominator is > 0 and is finite. Then {gi : i E J} is a partition of unity subordinate to the given atlas A. D Bibliography As general references and for collateral reading we suggest the following four books: [1] C. Chevalley: The Theory of Lie Groups 1. Princeton University Press, Princeton, 1946. A classic. Undoubtedly still the best in my opinion. [2] Y. Matsushima: Differential Manifolds. Marcel Dekker, Inc. New York 1972. [3] M. Spivak: A Comprehensive Introduction to Differential Geometry. Vol. I, Publish or Perish, Boston, 1970. [4] F. Warner: Foundations of Differential Manifolds and Lie Groups. Springer-Verlag 1984. For Lie groups, Lie algebras and representation theory we recommend: [5] V. S. Varadarajan: Lie Groups, Lie algebras and their Representa• tions. Springer-Verlag, GTM [6] A. W. Knapp: Lie Groups, Lie Algebras, and Cohomology. Mathe• matical Notes 34, Princeton University Press, Princeton, 1988. [7] A. W. Knapp: Representation Theory of Semisimple Lie Groups• An overview based on examples. Princeton University Press, Prince• ton, 1986. [8] S. Helgason: Differential Geometry, Lie groups and Symmetric Spaces. Vol I, Academic Press, New York, 1980. [9] S. Helagason: Groups and Geometric Analysis. Academic Press, 1984. BIBLIOGRAPHY 289 [10] N. Wallach: Harmonic Analysis on Homogeneous Spaces. Marcel Dekker Inc. New York, 1973. [11] N. Wallach: Real Reductive Groups 1. Academic Press, 1988. For a graded introduction to Differential and Riemannian Geometry, we highly recommend the following books in the order in which they appear: [12] M. P. do Carmo: Differential Geometry of Curves and Surfaces. Prentice-Hall, Engelwood, NJ, 1976 [13] A. Gray: Modem Differential Geometry of Curves and Surfaces with MATHEMATICA®. CRC Press, 1998. [14] W. Klingenberg: A Course in Differential Geometry. Springer• Verlag, 1978. [15] N. J. Hicks: Notes on Differential Geometry. D. Van Nostrand Com• pany Inc. Princeton, NJ 1965 [16] M.P. do Carmo: Riemannian Geometry. Birkhiiuser, Boston, 1992. [17] P. Petersen: Riemannian Geometry, GTM-I71, Springer-Verlag, 1997. [18] B. O'Neill: Semi-Riemannian Geometry. Academic Press, New York 1983. [19] M. Spivak: A Comprehensive Introduction to Differential Geometry. Vols. I-V, Publish or Perish, Boston, 1970. [20] S. Kobayashi and K. Nomizu: Foundations of Differential Geome• try. 2 Vols., 1963, 1969. [21] J. Cheeger & D. G. Ebin: Comparison Theorems in Riemannian Geometry. North-Holland Publishing Company, Amsterdam, 1975. A few other books mentioned in the book for reference. [22] J.L. Dupont: Curvature and Characteristic Classes, Lecture Notes in Mathematics, No. 640, Springer-Verlag, 1978 . [23] W. Massey: Basic Course in Algebraic Topology, GTM-127, Springer-Verlag, 1991. List of Symbols and Notation Ad(g), 150 V:(p), 172 ad(X), 150 ds2 , 242 A g ,80 Dvf(x),21 Altk,173 ~ 266 BL(E, F), 4 E,F,G,240 [X, Y], 110 eX, 31 B(x, r), 2 exp(X), 31, 132 C OO (M),76 Expx(p), 132 C k ,48 r(tp),188 0, 189 f.(X), 189 DOt, 52 A"Y, 172 df,184 rfj,251 df(p),95 gij, 239 Df(x),6 GL(E),48 Div (X), 229 GL(n, lR), 12 Vl(M), V 1(M), 184 gradf(x),l0 V(M), 189 Gr(r, V), 74 dw, 187,191 g,117 a~i 81 H(p),257 291 IMT, 32 R(Xp, Yp), R(x, y), 264 ix, 206 R(X, Y)Z, 264 K(p),257 Sk(t), 171 K p(P),277 SL(n, lR), 80 La, 80 sn,66 .cxw, 187 sn(R),68 Lie(G),117 SU(n),80 .cx(Y),110 sl(n, lR), 112 L k(E,F),48 su(n),112 L,M,N, 256 suppw, 215 f(-y), 242 Sn, 69 II II, 1 T")'(to), 268 II 11 2 , II 11 00 ,2 ®kV, ®V, 169 M(n,lR),7 Tp(n),81 \1 f(x), 40 T;M,95 nf,274 U(n), 80 O(n,lR),80 u(n), 112 o(g),5 V®W, 166 O(n),69 I\V,174 o( n, JR), 112 wf, 273 IF (lR), 73 X(p), X p , 109 R a,80 XU),109 R{kl' 274 X(M),109 Index Cl-diffeomorphism, 44 affine, 236 Djdt, 265 forms, 273 Exp-map, 132 Levi-Civita, 249 292 INDEX 293 Determinant, 179, 180 Group action, 155 Diffeomorphism, 79 transitive, 156 local,96 Differential Haar measure, 218 I-form, 95, 184 Hessian manifold, 64 of a function, 54 structure, 69 Homogeneous space, 156 system, 136 Hyperbolic metric, 241 Differential system, 136 integrable, 137 Imbedding, 101 involutive, 137 regular, 10 1 Directional derivative operator, Immersion, 44, 96 82 IMT,32 Divergence of a vector field, 229 Integral curve, 125 maximal, 128 Equations Integral submanifold, 137 Cauchy-Riemann, 14 Isometry, 246 Exponential map local,246 of Lie groups, 132 Isotropy subgroup, 160 of a vector field, 132 Exterior Jacobi identity, 112 algebra, 177 Jacobian of a map, 92 differentiation, 190 product, 174 Lagrange multiplier, 104 Extremum with constraints, 104 Level sets as manifolds, 69 Lie Frechet differentiable, 6 algebra of a Lie group, 117 Frobenius theorem, 110 algebra, 112 Function algebra, abelian, 112 coordinate, 76 bracket, 110 Fundamental form derivative, 110, 187, 203 first, 242 group, 80 second,256 Local Fundamental theorem maximum, 53 of calculus, 47 minimum, 53 Local one-parameter family, 130 Gauss equation, 254 Manifold map, 255 C k ,65 theorem, 260 00,72 Geodesics, 267 dimension of a, 66 Germs, 90 eight, 72 294 INDEX flag, 74 Product rule, 49 Grassmann, 74 Projective space, 73 level set as a, 69 orientable, 208 Rank Matrix of a tensor, contravariant, negative definite, 54 183 positive definite, 54 of a tensor, covariant, 187 Maximum contravariant, 187 local, 53 Regular Milnor's proof, 108, 120 imbedding, 101 Minimum submanifold, 101 local, 53 value, 101 Multilinear map, 165 Riemannian metric, 238 Multiplication Riemannian metrics interior, 206 conformally equivalent, 278 Norm, 1 Saddle point, 54 Loo,2 Section L 1 ,2 local, 159 equivalent, 2 Smooth function, 53, 75 Euclidean, 2 Smooth maps, 77 max, 2 Sphere, 66 of operator, 4 Spherical metric, 241 sup, 4 Step function, 46 uniform, 4 integral of a, 46 Stereographic projection, 66 One-parameter subgroup, 133 Submanifold, 100 One-parameter group, 130 integral, 137 Orientable manifold, 208 regular, 101 Orientation Submersion, 44, 96 boundary, 226 Support of a function, 55 of a vector space, 207 preserving, 213 Tangent preserving maps, 214 to a curve, 81 Oriented vector, 9 area, 173 vector to manifold, 85 vector space, 207 Taylor's theorem, 50 volume, 173 Tensor, 170 Orthogonal group, 106 algebra, 170 alternating, 172 Parallel transport, 234, 268 contravariant, 172 Poincare metric, 241 covariant, 95 INDEX 295 covariant rank of a, 183 product, 166 pull-back of a covariant, 188 rank, 172 skew-symmetric, 172 symmetric, 170 Theorem Brouwer's, 124 Frobenius, 110 Gauss, 260 Stokes, 226 Tayior's, 50 Transformation law for tensors, 171 rank 1 tensors, 89 Translation left, 81 right, 81 Vector normal,40 tangent, 9 Vector field, 106 ip-related, 114 along a curve, 265 complete, 128 divergence, 229 exponential map of a, 132 flow of a, 128 left invariant, 114 parallel along a curve, 267 Vector fields right invariant, 115 Volume oriented, 43, 173 Volume form of a Riemannian manifold, 222 Wedge product, 174 Weingarten map, 254