CHAPTER I. Manifolds and Vector Fields separated in one chart if TT(X) = ir(Y). So TM is again a smooth manifold in a canonical way the triple (TM,TTM,M) is called the tangent bundle of the manifold M. 1.10. Kinematic definition of the tangent space. Let Co°(R,M) de- note the space of germs at 0 of smooth curves R — • M. We put the following equivalence relation on Co°(R, M)\ the germ of c is equivalent to the germ of e if and only if c(0) = e(0) and in one (equivalently: each) chart (U,u) with c(0) = e(0) G U we have ^\Q(U O c)(t) = j||o(^ ° e)(i). The equiva- lence classes are also called velocity vectors of curves in M. We have the following diagram of mappings where a(c)(germc(0)/) = ^|o/(c(t)) and P : TM -+ C^°(M,M) is given by: /?((Tu)-1(y,y)) is the germ at 0 of t i— • u~1(y + tY). So TM is canonically identified with the set of all possible velocity vectors of curves in M: CHR.M)/- * C§ 0 0 / TTM 1.11. Tangent mappings. Let / : M — • A T be a smooth mapping between manifolds. Then / induces a linear mapping Txf :TXM — Tf^N for each x G M by (Txf.Xx)(h) = Xx(h o / ) for /i G C^ } (iV,E). This mapping is well defined and linear since /* : Cftx)(N,R) - • C£°(M,R), given by /i i— h o / , is linear and an algebra homomorphism, and T x / is its adjoint, restricted to the subspace of derivations. If (U,u) is a chart around x and (V,v) is one around f(x), then (T x f.&\ x ){vi) = £\ x (vjof) = ^ o / o O M ) , r*/.&|* = E,(^/-£d*)(^)&l/(*) by (1.8) So the matrix of Txf : T^M - T/(x)JV in the bases (fyx) and (^j|/(x)) is just the Jacobi matrix d(v o / o tt-1)(w(x)) of the mapping v o f o u~l at tt(x), so r/(a.)U o T x / o (T^u) - 1 = d(v o f o i r 1 ) ^ ^ ) ) . Let us denote by T / : TM —• TN the total mapping which is given by Tf\TxM := Txf. Then the composition TvoTfo (Tu)~l : u(U) x l r a ^ v(V) x Rn, (y, Y)~((vofo u-^iy), d(v o / o U -l){y)Y), is smooth thus Tf : T M — TN is again smooth.

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