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.
