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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