I want to use the theorem that states that if C is the set of critical points in N of a smooth function f:N->M (where N and M are manifolds), then f(C) is of measure zero in M.(adsbygoogle = window.adsbygoogle || []).push({});

in solving the next statement:

"if N is a submanifold of M, with dim(N)<dim(M), then N has measure zero in M".

Now let's look at: f(x)=x for f:N->M, this function is smooth, but because the differential of f, df isn't onto the tangent space of M, each value of N is critical, thus f(C)=N, so N is of measure zero in M by sard's theorem.

The book I'm using asks to prove this without sard's theorem which looks a bit hard, although I think I can assume wlog, that M=R^m and N=R^n, and n<m, but then I'm not sure how to prove that R^n has measure zero in R^m, without using the fact that R^n is diffeoemorphic to {(x1,...,xn,0,....,0) in R^m}.

any hints?

thanks in advance.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Implementing Sard's theorem.

**Physics Forums | Science Articles, Homework Help, Discussion**