Sard’s theorem
Let $\varphi :{X}^{n}\to {Y}^{m}$ be a smooth map on smooth manifolds^{}. A critical point^{} of $\varphi $ is a point $p\in X$ such that the differential^{} ${\varphi}_{*}:{T}_{p}X\to {T}_{\varphi (p)}Y$ considered as a linear transformation of real vector spaces has rank (http://planetmath.org/RankLinearMapping) $$. A critical value of $\varphi $ is the image of a critical point. A regular value of $\varphi $ is a point $q\in Y$ which is not the image of any critical point. In particular, $q$ is a regular value of $\varphi $ if $q\in Y\setminus \varphi (X)$.
Following Spivak [Spivak], we say a subset $V$ of ${Y}^{m}$ has measure zero if there is a sequence of coordinate charts $({x}_{i},{U}_{i})$ whose union contains $V$ and such that ${x}_{i}({U}_{i}\cap V)$ has measure 0 (in the usual sense) in ${\mathbb{R}}^{m}$ for all $i$. With that definition, we can now state:
Sard’s Theorem. Let $\varphi :X\to Y$ be a smooth map on smooth manifolds. Then the set of critical values of $\varphi $ has measure zero.
References
- 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 |