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
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