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MathematicalPhysicist
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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.
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.
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.