PlanetMath: Sard's theorem

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Sard's theorem

(Theorem)

Let $\phi : X^n \rightarrow Y^m$ be a smooth map on smooth manifolds. A critical point of $\phi$ is a point $p\in X$ such that the differential $\phi_* : T_pX \rightarrow T_{\phi(p)}Y$ considered as a linear transformation of real vector spaces has rank . 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, is a regular value of $\phi$ if $q\in Y \setminus \phi(X)$ .

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 $x_i(U_i\cap V)$ has measure 0 (in the usual sense) in $\mathbb{R}^m$ for all . 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.

Bibliography

Spivak: Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Volume I, Third Edition. Publish of Perish, Inc. Houston, Texas. 1999.

"Sard's theorem" is owned by mathcam. [ full author list (2) | owner history (1) ]

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See Also: residual, Baire category theorem

Also defines:

critical point, critical value, regular value

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Cross-references: measure, contains, union, coordinate charts, sequence, measure zero, subset, image, vector spaces, real, linear transformation, point, smooth manifolds, smooth map
There are 25 references to this entry.

This is version 6 of Sard's theorem, born on 2002-09-27, modified 2006-07-26.
Object id is 3477, canonical name is SardsTheorem.
Accessed 13546 times total.

Classification:

AMS MSC:

57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings)

Pending Errata and Addenda

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