Sard’s theorem

Let ϕ:XnYm be a smooth map on smooth manifoldsMathworldPlanetmath. A critical pointMathworldPlanetmath of ϕ is a point pX such that the differentialMathworldPlanetmath ϕ*:TpXTϕ(p)Y considered as a linear transformation of real vector spaces has rank ( <m. A critical value of ϕ is the image of a critical point. A regular value of ϕ is a point qY which is not the image of any critical point. In particular, q is a regular value of ϕ if qYϕ(X).

Following Spivak [Spivak], we say a subset V of Ym has measure zero if there is a sequence of coordinate charts (xi,Ui) whose union contains V and such that xi(UiV) has measure 0 (in the usual sense) in m for all i. With that definition, we can now state:

Sard’s Theorem. Let ϕ:XY be a smooth map on smooth manifolds. Then the set of critical values of ϕ has measure zero.


  • Spivak Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Volume I, Third Edition. Publish of Perish, Inc. Houston, Texas. 1999.
Title Sard’s theorem
Canonical name SardsTheorem
Date of creation 2013-03-22 13:04:09
Last modified on 2013-03-22 13:04:09
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 9
Author mathcam (2727)
Entry type Theorem
Classification msc 57R35
Related topic Residual
Related topic BaireCategoryTheorem
Defines critical point
Defines critical value
Defines regular value