Can Sard's Theorem Confirm Submanifolds as Measure Zero?

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SUMMARY

The discussion centers on Sard's Theorem, which asserts that the image of critical points of a smooth function between manifolds has measure zero. The user seeks to apply this theorem to demonstrate that a submanifold \( N \) of dimension less than that of manifold \( M \) has measure zero in \( M \). The function \( f(x) = x \) is analyzed, revealing that since its differential \( df \) is not onto the tangent space of \( M \), all points in \( N \) are critical, confirming that \( N \) is indeed of measure zero in \( M \) via Sard's Theorem. The user also explores proving this without relying on Sard's Theorem, specifically questioning how to establish that \( \mathbb{R}^n \) has measure zero in \( \mathbb{R}^m \).

PREREQUISITES
  • Understanding of Sard's Theorem in differential geometry
  • Familiarity with smooth functions between manifolds
  • Knowledge of measure theory and concepts of measure zero
  • Basic principles of submanifolds and their dimensions
NEXT STEPS
  • Study the proof of Sard's Theorem and its implications in differential geometry
  • Research measure theory, focusing on the properties of measure zero sets
  • Explore the concept of diffeomorphism and its role in manifold theory
  • Investigate alternative proofs for the measure zero property of submanifolds
USEFUL FOR

Mathematicians, particularly those specializing in differential geometry and measure theory, as well as students seeking to deepen their understanding of submanifolds and their properties in relation to higher-dimensional manifolds.

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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.
 
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