4 CHAPTER L Manifolds and Vector Fields

1.4. Smooth functions. The set of smooth real valued functions on a

manifold M will be denoted by C°°(M), in order to distinguish it clearly

from spaces of sections which will appear later. The space C°°(M) is a real

commutative algebra.

The support of a smooth function / is the closure of the set where it does not

vanish, supp(/) = {x G M : f(x) = £ 0}. The zero set of / is the set where /

vanishes, Z(f) = {x G M : f(x) = 0}.

1.5. Theorem. Any (separable, metrizable, smooth) manifold admits

smooth partitions of unity: Let {Ua)aeA be an open cover of M.

Then there is a family (ipa)aeA of smooth functions on M, such that:

(1) pa(x) 0 for all x G M and all a G A.

(2) supp((^a) C Ua for all a G A.

(3) (supp((pa))aeA ^ & locally finite family (so each x G M has an open

neighborhood which meets only finitely many supp((^a)y).

(4) ^2a ipa = 1 (locally this is a finite sum).

Proof. Any (separable, metrizable) manifold is a 'Lindelof space\ i.e., each

open cover admits a countable subcover. This can be seen as follows:

Let U be an open cover of M. Since M is separable, there is a countable

dense subset S in M. Choose a metric on M. For each U G U and each

x G U there is a y G S and n G N such that the ball Bi/n(y) with respect

to that metric with center y and radius ^ contains x and is contained in

U. But there are only countably many of these balls; for each of them we

choose an open set U G U containing it. This is then a countable subcover

ofW.

Now let (Ua)aeA be the given cover. Let us fix first a and x G Ua. We

choose a chart (U,u) centered at x (i.e., u(x) = 0) and e 0 such that

sBn

C u(UnUa), where

Dn

= {y G

W1

: \y\ 1} is the closed unit ball. Let

f e"

1

/' for t 0,

hit) :=

[0 for t 0,

a smooth function on R. Then

_ Jfe(e

2

-|«(*)|

2

) for^Gt/,

/ a M

*

J : _

\0 for zi U

is a nonnegative smooth function on M with support in Ua which is positive

at x.

We choose such a function fa^x for each a and x G Ua. The interiors of the

supports of these smooth functions form an open cover of M which refines