Let be a smooth map on smooth manifolds. A critical point of is a point such that the differential considered as a linear transformation of real vector spaces has rank (http://planetmath.org/RankLinearMapping) . A critical value of is the image of a critical point. A regular value of is a point which is not the image of any critical point. In particular, is a regular value of if .
Following Spivak [Spivak], we say a subset of has measure zero if there is a sequence of coordinate charts whose union contains and such that has measure 0 (in the usual sense) in for all . With that definition, we can now state:
Sard’s Theorem. Let be a smooth map on smooth manifolds. Then the set of critical values of has measure zero.
- Spivak Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Volume I, Third Edition. Publish of Perish, Inc. Houston, Texas. 1999.
|Date of creation||2013-03-22 13:04:09|
|Last modified on||2013-03-22 13:04:09|
|Last modified by||mathcam (2727)|