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

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 $X\subset \mathbb {R} ^{n}$ 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 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,^[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 $f:N\rightarrow M$ 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.^[4] In particular the Hausdorff dimension of $f(A_{r})$ is at most r. Caveat: The Hausdorff dimension of $f(A_{r})$ can be arbitrarily close to r.^[5]