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, let

:$f\colon \mathbb^n \rightarrow \mathbb^m$

be $C^k$, (that is, $k$ times continuously differentiable), where $k\geq \max\$. Let $X \subset \mathbb R^n$ denote the ''critical set'' of $f,$ which is the set of points $x\in \mathbb^n$ at which the Jacobian matrix of $f$ has rank . Then the image $f(X)$ has Lebesgue measure 0 in $\mathbb^m$.

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 $\mathbb^n$, it must have few critical ''values'' in the image $\mathbb^m$.

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\$, 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, and the general case by Arthur Sard in 1942.

A version for infinite-dimensional Banach manifolds was proven by Stephen Smale.

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\$ 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. 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''.

See also

* Generic property

References

Further reading

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{{Measure theory

Lemmas in analysis
Smooth functions
Multivariable calculus
Singularity theory
Theorems in analysis
Theorems in differential geometry
Theorems in measure theory