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Sard's theorem
Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis which asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set, i.e., it has Lebesgue measure 0. This makes the set of critical values "small" in the sense of a generic property. The theorem is named for Anthony Morse and Arthur Sard.
Statement
More explicitly (Sternberg (1964, Theorem II.3.1); Sard (1942)), let
\( f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m \)
be \( C^k, (that is, k times continuously differentiable), where \( k\geq \max\{n-m+1, 1\} \). Let X denote the critical set of f, which is the set of points \(x\in \mathbb{R}^n at which the Jacobian matrix of f has rank < m. Then the image f(X) has Lebesgue measure 0 in \( \mathbb{R}^m \).
Intuitively speaking, this means that although X may be large, its image must be small in the sense of Lebesgue measure: while f may have many critical points in the domain \( \mathbb{R}^n \), it must have few critical values in the image \( \mathbb{R}^m \).
More generally, the result also holds for mappings between second countable differentiable manifolds M and N of dimensions m and n, respectively. The critical set X of a \( C^k \)function
\( f:N\rightarrow M \)
consists of those points at which the differential
\( df:TN\rightarrow TM \)
has rank less than m as a linear transformation. If \( k\geq \max\{n-m+1,1\} \), then Sard's theorem asserts that the image of X has measure zero as a subset of M. This formulation of the result follows from the version for Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since a countable union of sets of measure zero is a set of measure zero, and the property of a subset of a coordinate patch having zero measure is invariant under diffeomorphism.
Variants
There are many variants of this lemma, which plays a basic role in singularity theory among other fields. The case m=1 was proven by Anthony P. Morse in 1939 (Morse 1939), and the general case by Arthur Sard in 1942 (Sard 1942).
A version for infinite-dimensional Banach manifolds was proven by Stephen Smale (Smale 1965).
The statement is quite powerful, and the proof is involved analysis. In topology it is often quoted — as in the Brouwer fixed point theorem and some applications in Morse theory — in order to use the weaker corollary that “a non-constant smooth map has a regular value”, and sometimes “...hence also a regular point”.
In 1965 Sard further generalized his theorem to state that if \( f:M\rightarrow N\) is \)C^k for k\geq \max\{n-m+1, 1\} \) and if \( A_r\subseteq M \)is the set of points \( x\in M \) such that \( df_x \) has rank strictly less than r, then the r-dimensional Hausdorff measure of \( f(A_r) \)is zero. In particular the Hausdorff dimension of \( f(A_r) \)is at most r. Caveat: The Hausdorff dimension of \( f(A_r) \)can be arbitrarly close to r.[1]
See also
Generic property
References
http://math.stackexchange.com/a/446049/62443
Sternberg, Shlomo (1964), Lectures on differential geometry, Englewood Cliffs, NJ: Prentice-Hall, pp. xv+390, MR 0193578, Zbl 0129.13102.
Morse, Anthony P. (January 1939), "The behaviour of a function on its critical set", Annals of Mathematics 40 (1): 62–70, doi:10.2307/1968544, JSTOR 1968544, MR 1503449.
Sard, Arthur (1942), "The measure of the critical values of differentiable maps", Bulletin of the American Mathematical Society 48 (12): 883–890, doi:10.1090/S0002-9904-1942-07811-6, MR 0007523, Zbl 0063.06720.
Sard, Arthur (1965), "Hausdorff Measure of Critical Images on Banach Manifolds", American Journal of Mathematics 87 (1): 158–174, doi:10.2307/2373229, JSTOR 2373229, MR 0173748, Zbl 0137.42501 and also Sard, Arthur (1965), "Errata to Hausdorff measures of critical images on Banach manifolds", American Journal of Mathematics 87 (3): 158–174, doi:10.2307/2373229, JSTOR 2373074, MR 0180649, Zbl 0137.42501.
Smale, Stephen (1965), "An Infinite Dimensional Version of Sard's Theorem", American Journal of Mathematics 87 (4): 861–866, doi:10.2307/2373250, JSTOR 2373250, MR 0185604, Zbl 0143.35301.
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