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,{{citation | first=Arthur | last=Sard | author-link=Arthur Sard | title=The measure of the critical values of differentiable maps | url=http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07811-6/home.html | journal=Bulletin of the American Mathematical Society | volume=48 | year=1942 | issue=12 | pages=883–890 | mr= 0007523 | zbl= 0063.06720 | doi=10.1090/S0002-9904-1942-07811-6 |postscript=. | doi-access=free}} 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 $. Then the image f(X) has Lebesgue measure 0 in \mathbb{R}^m .$

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,{{citation | first= Anthony P. | last=Morse | author-link = Anthony Morse | title=The behaviour of a function on its critical set | journal=Annals of Mathematics | volume=40 | issue=1 |date=January 1939 | pages=62–70 | jstor=1968544 | doi=10.2307/1968544 | bibcode=1939AnMat..40...62M | mr=1503449 |postscript=.}} and the general case by Arthur Sard in 1942.

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale.{{citation | first=Stephen | last=Smale | author-link=Stephen Smale | title=An Infinite Dimensional Version of Sard's Theorem | journal=American Journal of Mathematics | volume=87 | year=1965 | pages=861–866 | jstor= 2373250 | doi=10.2307/2373250 | issue=4 | mr=0185604 | zbl=0143.35301 |postscript=. }}

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 $C^\infty$ and if $A_r\subseteq N$ is the set of points $x\in N$ such that $df_x$ has rank less or equal than $r$ , then the Hausdorff dimension of $f(A_r)$ is at most $r$ .{{citation | first=Arthur | last=Sard | title=Hausdorff Measure of Critical Images on Banach Manifolds | journal=American Journal of Mathematics | volume=87 | year=1965 | pages=158–174 | doi=10.2307/2373229 | issue=1 | jstor=2373229 | mr=0173748 | zbl=0137.42501 }} and also {{Citation | first=Arthur | last=Sard | title = Errata to Hausdorff measures of critical images on Banach manifolds | journal=American Journal of Mathematics | volume=87 | year=1965 | pages=158–174 | issue=3 | jstor = 2373074 | doi = 10.2307/2373229 | mr = 0180649 | zbl = 0137.42501 |postscript=. }}{{citation |title=Show that f(C) has Hausdorff dimension at most zero |date=July 18, 2013 |work=Stack Exchange |url=https://math.stackexchange.com/q/446049 }}

Sard's theorem

Statement

Variants

See also

References

Further reading