Sard's theorem - Imported Knowlpedia Content

Statement

Let $M$ and $N$ be finite-dimensional, second-countable smooth manifolds without boundary, and let $f:M\to N$ be a smooth map. Sard's theorem states that the set of critical values of $f$ has measure zero in $N$ : in every smooth coordinate chart of $N$ , its image has Lebesgue measure zero. Consequently, almost every $y\in N$ is a regular value of $f$ . No measure or volume form is part of the data; the measure-zero conclusion is invariant under smooth coordinate changes.

Differentiability threshold

The smooth hypothesis can be weakened. If $\dim M=m$ , $\dim N=n$ , and $f$ is $C^k$ , the Morse-Sard theorem holds when

k>\max\{m-n,0\}.

This threshold is essential in general; lower-regularity maps can have critical value sets of positive measure. The precise finite-differentiability statement is given in Lee, Chapter 6.

Consequences and examples

If the target is nonempty, regular values are dense because a measure-zero subset cannot contain a coordinate-open set. Combined with the regular-level set theorem, Sard's theorem produces regular fibers and is the measure-theoretic engine behind generic transversality arguments.

For a constant map $M\to N$ with $\dim N>0$ , its single image point is a critical value and has measure zero. The theorem does not say that the set of critical points in $M$ is small.

Conventions and scope

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: Chapter 6, Sard's theorem and regular values.
Victor Guillemin and Alan Pollack, Differential Topology, AMS Chelsea Publishing, 2010 reprint. DOI record. Relevant: Chapter 1, Sard's theorem and its applications.