Sard's theorem

Short description: Theorem in mathematical analysis

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

[math]\displaystyle{ f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m }[/math]

be [math]\displaystyle{ C^k }[/math], (that is, [math]\displaystyle{ k }[/math] times continuously differentiable), where [math]\displaystyle{ k\geq \max\{n-m+1, 1\} }[/math]. Let [math]\displaystyle{ X \subset \mathbb R^n }[/math] denote the critical set of [math]\displaystyle{ f, }[/math] which is the set of points [math]\displaystyle{ x\in \mathbb{R}^n }[/math] at which the Jacobian matrix of [math]\displaystyle{ f }[/math] has rank [math]\displaystyle{ \lt m }[/math]. Then the image [math]\displaystyle{ f(X) }[/math] has Lebesgue measure 0 in [math]\displaystyle{ \mathbb{R}^m }[/math].

Intuitively speaking, this means that although [math]\displaystyle{ X }[/math] may be large, its image must be small in the sense of Lebesgue measure: while [math]\displaystyle{ f }[/math] may have many critical points in the domain [math]\displaystyle{ \mathbb{R}^n }[/math], it must have few critical values in the image [math]\displaystyle{ \mathbb{R}^m }[/math].

More generally, the result also holds for mappings between differentiable manifolds [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] of dimensions [math]\displaystyle{ m }[/math] and [math]\displaystyle{ n }[/math], respectively. The critical set [math]\displaystyle{ X }[/math] of a [math]\displaystyle{ C^k }[/math] function

[math]\displaystyle{ f:N\rightarrow M }[/math]

consists of those points at which the differential

[math]\displaystyle{ df:TN\rightarrow TM }[/math]

has rank less than [math]\displaystyle{ m }[/math] as a linear transformation. If [math]\displaystyle{ k\geq \max\{n-m+1,1\} }[/math], then Sard's theorem asserts that the image of [math]\displaystyle{ X }[/math] has measure zero as a subset of [math]\displaystyle{ M }[/math]. 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 [math]\displaystyle{ m=1 }[/math] 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 [math]\displaystyle{ f:N\rightarrow M }[/math] is [math]\displaystyle{ C^k }[/math] for [math]\displaystyle{ k\geq \max\{n-m+1, 1\} }[/math] and if [math]\displaystyle{ A_r\subseteq N }[/math] is the set of points [math]\displaystyle{ x\in N }[/math] such that [math]\displaystyle{ df_x }[/math] has rank strictly less than [math]\displaystyle{ r }[/math], then the r-dimensional Hausdorff measure of [math]\displaystyle{ f(A_r) }[/math] is zero. In particular the Hausdorff dimension of [math]\displaystyle{ f(A_r) }[/math] is at most r. Caveat: The Hausdorff dimension of [math]\displaystyle{ f(A_r) }[/math] can be arbitrarily close to r.

See also

• Generic property

Further reading

• Hirsch, Morris W. (1976), Differential Topology, New York: Springer, pp. 67–84, ISBN 0-387-90148-5. 
• Sternberg, Shlomo (1964), Lectures on Differential Geometry, Englewood Cliffs, NJ: Prentice-Hall.