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