Sard's theorem

In mathematics, 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,^[1] let

${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} ^{m}}$

be ${\displaystyle C^{k}}$, (that is, ${\displaystyle k}$ times continuously differentiable), where ${\displaystyle k\geq \max\{n-m+1,1\}}$. Let ${\displaystyle X\subset \mathbb {R} ^{n}}$ denote the critical set of ${\displaystyle f,}$ which is the set of points ${\displaystyle x\in \mathbb {R} ^{n}}$ at which the Jacobian matrix of ${\displaystyle f}$ has rank ${\displaystyle <m}$. Then the image ${\displaystyle f(X)}$ has Lebesgue measure 0 in ${\displaystyle \mathbb {R} ^{m}}$.

Intuitively speaking, this means that although ${\displaystyle X}$ may be large, its image must be small in the sense of Lebesgue measure: while ${\displaystyle f}$ may have many critical points in the domain ${\displaystyle \mathbb {R} ^{n}}$, it must have few critical values in the image ${\displaystyle \mathbb {R} ^{m}}$.

More generally, the result also holds for mappings between differentiable manifolds ${\displaystyle M}$ and ${\displaystyle N}$ of dimensions ${\displaystyle m}$ and ${\displaystyle n}$, respectively. The critical set ${\displaystyle X}$ of a ${\displaystyle C^{k}}$ function

${\displaystyle f:N\rightarrow M}$

consists of those points at which the differential

${\displaystyle df:TN\rightarrow TM}$

has rank less than ${\displaystyle m}$ as a linear transformation. If ${\displaystyle k\geq \max\{n-m+1,1\}}$, then Sard's theorem asserts that the image of ${\displaystyle X}$ has measure zero as a subset of ${\displaystyle M}$. 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 ${\displaystyle m=1}$ was proven by Anthony P. Morse in 1939,^[2] and the general case by Arthur Sard in 1942.^[1]

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale.^[3]

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 ${\displaystyle f:N\rightarrow M}$ is ${\displaystyle C^{k}}$ for ${\displaystyle k\geq \max\{n-m+1,1\}}$ and if ${\displaystyle A_{r}\subseteq N}$ is the set of points ${\displaystyle x\in N}$ such that ${\displaystyle df_{x}}$ has rank strictly less than ${\displaystyle r}$, then the r-dimensional Hausdorff measure of ${\displaystyle f(A_{r})}$ is zero.^[4] In particular the Hausdorff dimension of ${\displaystyle f(A_{r})}$ is at most r. Caveat: The Hausdorff dimension of ${\displaystyle f(A_{r})}$ can be arbitrarily close to r.^[5]

See also

• Generic property

References

1. 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.
2. Morse, Anthony P. (January 1939), "The behaviour of a function on its critical set", Annals of Mathematics, 40 (1): 62–70, Bibcode:1939AnMat..40...62M, doi:10.2307/1968544, JSTOR 1968544, MR 1503449.
3. 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.
4. 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.
5. "Show that f(C) has Hausdorff dimension at most zero", Stack Exchange, July 18, 2013

Further reading

• Hirsch, Morris W. (1976), Differential Topology, New York: Springer, pp. 67–84, ISBN 0-387-90148-5.
• Sternberg, Shlomo (1964), Lectures on Differential Geometry, Englewood Cliffs, NJ: Prentice-Hall, MR 0193578, Zbl 0129.13102.