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)/- *
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.
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