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
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