Sard's Theorem
Get Sard's Theorem essential facts below. View Videos or join the Sard's Theorem discussion. Add Sard's Theorem to your PopFlock.com topic list for future reference or share this resource on social media.
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 ${\displaystyle X}$ 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]

## References

1. ^ a b 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, 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