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Sard's theorem

Summary

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  . 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]