1. Differentiable Manifolds 5
(Ua)j so by the argument at the beginning of the proof there is a countable
subcover with corresponding functions /i , /2, Let
Wn = {xeM : fn(x) 0 and fi(x) ± for 1 i n},
and denote by Wn the closure. Then (Wn)n is an open cover. We claim
that (Wn)n is locally finite: Let x G M. Then there is a smallest n such
that x G Wn. Let V := {y G M : /n(j/) j/nO*)}. If i/ G V n Wk, then we
have /
n
(y) \fn{%)
a n
d /i(y) \ for i fc, which is possible for finitely
many k only.
Consider the nonnegative smooth function
9n(x) = h(fn(x))h(± - h(x)) . . . / * ( £ - /n-i(x)), n G N.
Then obviously supp(#n) = Wn. So p := J2n9n is smooth, since it is locally
only a finite sum, and everywhere positive; thus (gn/g)neN is a smooth
partition of unity on M. Since supp(gn) = Wn is contained in some f/a(n),
we may put tpa ^2sn-a(n)=a\ ~- to get the required partition of unity which
is subordinated to (Ua)aeA- D
1.6. Germs. Let M and N be manifolds and x G M. We consider all
smooth mappings f : Uf N, where Uf is some open neighborhood of
x in M, and we put / ~
x
g if there is some open neighborhood V of x
with /| V = g\V. This is an equivalence relation on the set of mappings
considered. The equivalence class of a mapping / is called the germ of f at
x, sometimes denoted by germx / . The set of all these germs is denoted by
C™(M,N).
Note that for a germs at x of a smooth mapping only the value at x is
defined. We may also consider composition of germs: germyvx\ gogernXp / :=
germ^flfo/).
If iV = R, we may add and multiply germs of smooth functions, so we get
the real commutative algebra C^°(M, R) of germs of smooth functions at x.
This construction works also for other types of functions like real analytic
or holomorphic ones if M has a real analytic or complex structure.
Using smooth partitions of unity (1.4) it is easily seen that each germ of a
smooth function has a representative which is defined on the whole of M.
For germs of real analytic or holomorphic functions this is not true. So
C£°(M,R) is the quotient of the algebra C°°(M) by the ideal of all smooth
functions / : M R which vanish on some neighborhood (depending on / )
of x.
1.7. The tangent space of
Rn.
Let a G
Rn.
A tangent vector with foot
point a is simply a pair (a, X) with X G
Rn,
also denoted by Xa. It induces
a derivation Xa :
C°°(Rn)
- R by XJf) = df(a)(Xa). The value depends
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