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.
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.