110 XVI DIFFERENTIAL MANIFOLDS

is z fibered chart on T(M) (16.15.1), called the fibered chart associated with
the chart c = (U, <p, ri) on M. If (U, <p', n) is another chart on M with the same
domain of definition, and if u = <p Q <p'~~l : <p'(U) -* <p(U) is the transition
diffeomorphism, then the transition diffeomorphism for the associated fibered
charts on T(M) is the mapping

(16.15.4.5) (*, H)I->(M(JC), DM(X) • h)

of <p'(U) x RM onto <p(U) x R".

If a vector field J on U is given by (16,15.4.3), its local expression relative
to the fibered chart associated with (U, <p, ri) (16.15.1.3) is the mapping

(16.15.4.6) *»

of <p(U) into <p(U) x R".

(16.15.5) When M is an open subset of Rw, the inverse of the framing as-
sociated with the chart (M, 1M, ri) is a trivtalization of T(M), called the can-
onical trivialization and given by hxt-+(x, ^(hj) (16.5.2). Usually we shall
identify T(M) with M x Rn by means of this trivialization. A vector field on
M is then of the form x\-~*(x, f(X)), where f is a mapping of M into Rn.

(16.15.6) Let M,N be two differential manifolds and let /: M -»N be a
mapping of class Cr (where r is an integer ^ 1, or -f oo). Then the mapping

(16.15.6.1) T(/):h^TOM(h)(/)-h

(which we shall write in the more legible form
(16.15.6.2)

is a mapping of class C"~ 1 (with the convention that r - 1 = oo if r = oo) of
T(M) into T(N). For if (U, q>9 m) and (V, ^, ri) are charts on M and N, re-
spectively, such that/(U) c V, and if F is the local expression of /relative to
these charts (16.3), then the local expression of T(/) relative to the associated
fibered charts (16.15.4) is the mapping

(16.15.6.3) (X, h) K» (F(*), F(x) • h)

of ^(U) x. Rm into>(V) x R". If/is of class C00, then (/, T(/)) is a morphism
of vector bundles (16.15.2). It is clear thatsufficient that the a1 should be of class Cr.