Sard's theorem explained

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 R ^{n} → R ^{m}

be

C ^{k} , (that is, k times continuously differentiable), where k\geq max \{n - m +1 , 1 \} . Let X \subset R ^{n} denote the critical set of f, which is the set of points x\in R ^{n} at which the Jacobian matrix of f has rank <m . Then the image f(X) has Lebesgue measure 0 in 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 R ^{n} , it must have few critical values in the image 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 df:TN → 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, 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 → M is C ^{k} for k\geq max \{n - m +1 , 1 \} 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.^[1] 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

Notes and References

1. and also