1. Differentiable Manifolds 13 The Baire theorem says that a residual subset of a complete metric space is dense. 1.18. Regular values. Let / : M T V be a smooth mapping between manifolds. (1) A point x G M is called a singular point of / if Txf is not surjective, and it is called a regular point of / if Txf is surjective. (2) A point y G N is called a regular value of / if Txf is surjective for all x G f~x(y) If not, y is called a singular value. Note that any y G N \ f(M) is a regular value. Theorem ([166], [196]). The set of all singular values of a Ck mapping f : M N is of Lebesgue measure 0 in N ifk max{0, dim(M)—dim(iV)}. So any smooth mapping has regular values. Proof. We proof this only for smooth mappings. It is sufficient to prove this locally. Thus we consider a smooth mapping / : U » W1 where U C W71 is open. If n ra, then the result follows from lemma (1.17) above (consider the set U x 0 C Rm x Rn~m of measure 0). Thus let ra n. Let £(/ ) C U denote the set of singular points of / . Let / = (/*,..., / n ) , and let £(/ ) = S i U S 2 U S 3 where: Ei is the set of singular points x such that Pf(x) = 0 for all linear differ- ential operators P of order ~ . £2 is the set of singular points x such that Pf(x) ^ 0 for some differential operator P of order 2. £3 is the set of singular points x such that ~^-(x) = 0 for some i, j . We first show that /(£i ) has measure 0. Let v = \^ + 1] be the smallest integer m/n. Then each point of £1 has an open neighborhood W C U such that \f(x) - f(y)\ K\x - y\y for all x G £1 H W and y eW and for some K 0, by Taylor expansion. We take W ^ to be a cube, of width w. It suffices to prove that jf(£i H W) has measure 0. We divide W into pm cubes of width ~ those which meet Si\ will be denoted by Ci,... , C9 for g p m . Each Ck is contained in a ball of radius ^y/rn centered at a point of £1D W. The set f(Ck) is contained in a cube C'k C W1 of width 2K(^y/m)iy. Then J2»n(C'k) pm(2K)n(~V^)I/n = pm~un{2K)nwvn - 0 for p - 00, since m vn 0.
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