12 CHAPTER I. Manifolds and Vector Fields

1.16. Corollary. (1) The (separable) connected smooth manifolds are ex-

actly the smooth retracts of connected open subsets of

Rn

's.

(2) A smooth mapping f : M — » N is an embedding of a submanifold if

and only if there is an open neighborhood U of f(M) in N and a smooth

mapping r : U — M with r o f = MM-

Proof. Any manifold M may be embedded into some

Mn;

see (1.19) below.

Then there exists a tubular neighborhood of M in

Rn

(see later or [84, pp.

109-118]), and M is clearly a retract of such a tubular neighborhood. The

converse follows from (1.15).

For the second assertion we repeat the argument for T V instead of

Mn.

•

1.17. Sets of Lebesque measure 0 in manifolds. An m-cube of width

w 0 in E

m

is a set of the form C = [#i, x\ + w] x ... x [xm, xm + w\.

The measure

/JL(C)

is then /i(C) =

wn.

A subset S C

Rm

is called a set of

(Lebesque) measure 0 if for each e 0 these are at most countably many

m-cubes d with S C I J £ o ^

anc^

X^SoM^)

£-

Obviously, a countable

union of sets of Lebesque measure 0 is again of measure 0.

Lemma. Let U C

Rm

be open and let f : U -

Rm

be

C1.

If S C U is of

measure 07 then also f(S) C

Mm

is of measure 0.

Proof. Every point of S belongs to an open ball B C U such that the

operator norm ||d/(x)|| KB for all x G B. Then \f(x) — f(y)\ KB\X — y\

for all x, y G B. So if C C B is an m-cube of width w, then /(C) is contained

in an m-cube C of width ^s/rnKBW and measure

Now let S = [JjLi Sj where each Sj is a compact subset of a ball Bj as

above. It suffices to show that each f(Sj) is of measure 0.

For each e 0 there are m-cubes Ci in Bj with Sj C (Ji @i

a n

d Yli M(Ci)

£:-

As we saw above, then /(X^) C (Ji C^ with ^ - /i(^)

mml2K^.e.

D

Let M be a smooth (separable) manifold. A subset 5 C M is called a se£

o/ (Lebesque) measure 0 if for each chart ({7, u) of M the set u(S fl J7) is of

measure 0 in E

m

. By the lemma it suffices that there is some atlas whose

charts have this property. Obviously, a countable union of sets of measure

0 in a manifold is again of measure 0.

An m-cube is not of measure 0. Thus a subset of

W71

of measure 0 does

not contain any m-cube; hence its interior is empty Thus a closed set of

measure 0 in a manifold is nowhere dense. More generally, let S be a subset

of a manifold which is of measure 0 and cr-compact, i.e., a countable union of

compact subsets. Then each of the latter is nowhere dense, so 5 is nowhere

dense by the Baire category theorem. The complement of S is residual,

i.e., it contains the intersection of a countable family of open dense subsets.