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 image of the set of critical points of a smooth function f from one Euclidean space or manifold to another has Lebesgue measure 0 – they form a null set. This makes it "small" in the sense of a generic property.

Statement

More explicitly (Sternberg (1964, Theorem II.3.1); Sard (1942)), let

$f\colon \mathbb{R}^n \rightarrow \mathbb{R}^m$

be C^k, k times continuously differentiable, where k ≥ max{n−m+1, 1}. Let X be the critical set of f, the set of points x in Rⁿ at which the Jacobian matrix of f has rank < m. Then f(X) has Lebesgue measure 0 in R^m.

Thus although X itself may be large, its image must be small: f may have many critical points (in the domain Rⁿ), it has few critical values (in the target space 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 → M consists of those points at which the differential df : TN → TM has rank less than m as a linear transformation. If k ≥ 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 → N is C^k for k ≥ max{n−m+1, 1} and if A_r ⊆ M is the set of points x ∈ M such that df_x has rank less than or equal to r then f(A_r) has Hausdorff dimension at most r.

See also

• Jacobian criterion / generic smoothness, similar statements in algebraic geometry

References

• Sternberg, Shlomo (1964), Lectures on differential geometry, Englewood Cliffs, NJ: Prentice-Hall, pp. xv+390, MR0193578, Zbl 0129.13102 .
• 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, MR1503449 .
• 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, MR0007523, Zbl 0063.06720, http://www.ams.org/bull/1942-48-12/S0002-9904-1942-07811-6/home.html .
• 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, MR0173748, 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, MR0180649, Zbl 0137.42501 .
• 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, MR0185604, Zbl 0143.35301 .