Difference between revisions of "Sard theorem"
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| − | + | [[Category:Global analysis]] | |
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| + | '''Theorem''' | ||
| + | Let $M$ and $N$ be two [[Differentiable manifold|$C^r$ manifolds]] and $f:M\to N$ a $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, then the [[Critical value|critical values]] of $f$ form a set of measure zero. Therefore the set of regular values (see [[Singularities of differentiable mappings]]) has full measure. | ||
| + | The theorem was proved by A. Sard in {{Cite|Sa}}. Observe that there is no uniquely defined measure on $N$ and the statement means that, if $S\subset N$ denotes the (closed) subset of singular values of $f$, then, for every chart $(U, \phi)$ in the atlas defining $N$, $\phi (U\cap S)$ is a set of (Lebesgue) measure zero. | ||
| + | As a corollary of Sard's theorem we conclude that the set of regular values is dense. Thus $S$ is a [[Meager set|meager set]]. The latter statement is also sometimes called ''Sard's theorem'': however it is not equivalent to the one above, since closed meager sets might have positive Lebesgue measure. | ||
| − | ==== | + | ====References==== |
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| + | |valign="top"|{{Ref|Sa}}|| A. Sard, "The measure of critical values of differentiable maps" ''Bull. Amer. Math. Soc.'' , '''48''' (1942) pp. 883–890 {{MR|7523}} {{ZBL|0063.06720}} | ||
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Latest revision as of 16:38, 17 November 2012
2020 Mathematics Subject Classification: Primary: 58A05 Secondary: 28A [MSN][ZBL]
Theorem Let $M$ and $N$ be two $C^r$ manifolds and $f:M\to N$ a $C^r$ map. If $r> \max \{0, \dim M - \dim N\}$, then the critical values of $f$ form a set of measure zero. Therefore the set of regular values (see Singularities of differentiable mappings) has full measure.
The theorem was proved by A. Sard in [Sa]. Observe that there is no uniquely defined measure on $N$ and the statement means that, if $S\subset N$ denotes the (closed) subset of singular values of $f$, then, for every chart $(U, \phi)$ in the atlas defining $N$, $\phi (U\cap S)$ is a set of (Lebesgue) measure zero.
As a corollary of Sard's theorem we conclude that the set of regular values is dense. Thus $S$ is a meager set. The latter statement is also sometimes called Sard's theorem: however it is not equivalent to the one above, since closed meager sets might have positive Lebesgue measure.
References
| [Sa] | A. Sard, "The measure of critical values of differentiable maps" Bull. Amer. Math. Soc. , 48 (1942) pp. 883–890 MR7523 Zbl 0063.06720 |
Sard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sard_theorem&oldid=11998