Sard's theorem

Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that 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 $<\min\{m,n\}$. 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 involves analysis. In topology it is often quoted — as in the Brouwer fixed point theorem and some applications in Morse theory — in order to prove the weaker corollary that “a non-constant smooth map has at least one regular value”.

In 1965 Sard further generalized his theorem to state that if $f:N\rightarrow M$ is $C^{k}$ for $k\geq \max\{n-m+1,1\}$ and if $A_{r}\subseteq N$ is the set of points $x\in N$ 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.

See also

• Generic property

References

• 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.