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

(Txf.&\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.