# 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]