# Sard’s theorem

Let $\phi:X^{n}\rightarrow Y^{m}$ be a smooth map on smooth manifolds  . A of $\phi$ is a point $p\in X$ such that the differential  $\phi_{*}:T_{p}X\rightarrow T_{\phi(p)}Y$ considered as a linear transformation of real vector spaces has rank (http://planetmath.org/RankLinearMapping) $. A critical value of $\phi$ is the image of a critical point. A regular value of $\phi$ is a point $q\in Y$ which is not the image of any critical point. In particular, $q$ is a regular value of $\phi$ if $q\in Y\setminus\phi(X)$.

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 $\phi:X\rightarrow Y$ be a smooth map on smooth manifolds. Then the set of critical values of $\phi$ 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 SardsTheorem 2013-03-22 13:04:09 2013-03-22 13:04:09 mathcam (2727) mathcam (2727) 9 mathcam (2727) Theorem msc 57R35 Residual BaireCategoryTheorem critical point critical value regular value