1. Differentiable Manifolds 7 If (17,u) is a chart on M with x G t / , then u* : f *- fou induces an isomor- phism of algebras C ^ (Rn, R) = C£°(M,R), and thus also an isomorphism Txu : TXM - Tu(a0Rn, given by (T^.X X )(/) = X x ( / o u). So TXM is an n-dimensional vector space. We will use the following notation: u = (ul,..., un), so u% denotes the i-th coordinate function on £/, and &\x •= (TxuyH&lvw) = (Txu)-l(u(x),ei). So -J=^ \x € TXM is the derivation given by d_\ t t \ - djfou-1) dxl Prom (1.7) we have now *?!*(/) = — s i s — M 1 ) ) - T^.X, = ^(T^.X,)^)^!^) = £*(** °u)^| tt(!r) n n X x = (T x u)-\T x u.X x = Y^X^u1)^. i=l 1.9. The tangent bundle. For a manifold M of dimension n we put TM := \_\xeM TXM, the disjoint union of all tangent spaces. This is a family of vector spaces parameterized by M, with projection -KM • TM — M given by irM(TxM) = x. For any chart (Ua,ua) of M consider the chart {TxJ^{Ua),Tua) on TM, where Tua : -Kj^(Ua) — ua(Ua) x Rn is given by Tu a .X = (^(TTMpO^T^pQT/a.X). Then the chart changings look as follows: Tup o (Tua)-1 : Tua(7r^(Ua(3)) = ua(Ua0) x Rn - • -v u ^ t / ^ ) x Mn = Tupi^iU^)), ({Tufio(Tua)-l)(v,Y))U) = ((Tua)-\y,Y)){foup) = (v,Y)(f o up o u'1) = d(foUpoU~1)(y).Y = df{up o u~l(y)).d(up o u~ 1 )(y).r = (n / 3 ot i - 1 (y),d(n / 3 o«- 1 )(y).y)(/). So the chart changings are smooth. We choose the topology on TM in such a way that all Tua become homeomorphisms. This is a Hausdorff topology, since X, Y E TM may be separated in M if 7r(X) ^ nCY) and they may be

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.