nLab Sard's theorem

Named after Anthony P. Morse and Arthur Sard.


Recall the following definitions from differential topology. The set of critical points of a smooth map ff is the set of points in the domain of ff where the tangent map is not surjective. The set of critical values of ff is the ff-image of the set of critical points of ff. The set of regular values of ff is the complement of the set of critical values of ff.

Suppose MM and NN are smooth manifolds of dimension mm and nn respectively and f:MNf\colon M\to N is a C r\mathrm{C}^r-smooth map, where r1r\ge1 and r>mnr\gt m-n. Then the set of critical values in NN is a meager subset (alias first category subset) and a negligible subset (alias measure zero subset) of NN. In particular, the set of regular values is dense in NN. Furthermore, the ff-image of points of MM where ff has rank at most rr (0<r<m0\lt r\lt m) has Hausdorff dimension at most rr.

If NN is a Banach manifold and q1q\ge1, ff is a Fredholm map, and qq is strictly greater than the index of ff, then the critical values of ff form a meager subset of NN.


The case n=1n=1:

  • Anthony P. Morse, The Behavior of a Function on Its Critical Set, Annals of Mathematics 40:1 (1939), 62–70. doi.

The case n>1n\gt1:

  • Arthur Sard, The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, doi.

The case when NN is a Banach manifold:

  • Stephen Smale, An Infinite Dimensional Version of Sard’s Theorem, American Journal of Mathematics 87:4 (1965), 861–866. doi.

The part concerning Hausdorff measures?:

  • Arthur Sard, Hausdorff Measure of Critical Images on Banach Manifolds, American Journal of Mathematics 87:1 (1965), 158–174. doi.

Last revised on July 17, 2022 at 22:26:23. See the history of this page for a list of all contributions to it.