Can Sard's Theorem Confirm Submanifolds as Measure Zero?

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In summary, the conversation discusses the use of a theorem stating that if C is the set of critical points in N of a smooth function f:N->M, then f(C) is of measure zero in M. The theorem is then applied to solve the statement "if N is a submanifold of M, with dim(N)<dim(M), then N has measure zero in M". The conversation also mentions the use of Sard's theorem to prove this statement, but the book asks for a proof without using Sard's theorem. The speaker suggests assuming WLOG that M=R^m and N=R^n, and proving that R^n has measure zero in R^m without using the fact that R^n is diffeomorphic to {(
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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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Can you prove that R has measure zero in R^2?
 

FAQ: Can Sard's Theorem Confirm Submanifolds as Measure Zero?

What is Sard's theorem?

Sard's theorem is a fundamental result in differential topology that states that the set of critical values of a smooth map between two manifolds has measure zero in the target manifold.

How is Sard's theorem used in mathematics?

Sard's theorem is used in various areas of mathematics, such as differential geometry, differential topology, and partial differential equations. It has many applications, including proving the existence of smooth functions with prescribed singularities and studying the regularity of solutions to partial differential equations.

Can you provide an example of Sard's theorem in action?

One example of Sard's theorem in action is the proof of the fundamental theorem of algebra. By considering the map from the complex plane to itself given by the polynomial function, we can use Sard's theorem to show that almost every complex number is a regular value of this map, which implies that the polynomial has a root.

What are the assumptions for Sard's theorem to hold?

Sard's theorem holds under the assumption that the map between the manifolds is smooth and the dimensions of the manifolds are compatible. For example, the target manifold should have a higher dimension than the source manifold.

Are there any generalizations of Sard's theorem?

Yes, there are several generalizations of Sard's theorem, such as the strong Sard's theorem, which states that the set of critical values of a map has measure zero in the target manifold even when the map is not smooth, but only continuous. There are also versions of Sard's theorem for non-smooth maps between different types of spaces, such as Banach spaces and Fréchet spaces.

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