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Difference between revisions of "Sard theorem"

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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832301.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832302.png" />-mapping of manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832304.png" /> of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832306.png" />, respectively; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832307.png" />, then the critical values (cf. [[Critical value|Critical value]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832308.png" /> form a set of measure zero. The set of regular values turns out to be of full measure and everywhere dense. The theorem was proved by A. Sard [[#References|[1]]].
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====References====
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[[Category:Global analysis]]
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Sard, "The measure of critical values of differentiable maps" ''Bull. Amer. Math. Soc.'' , '''48''' (1942) pp. 883–890 {{MR|7523}} {{ZBL|0063.06720}} </TD></TR></table>
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'''Theorem'''
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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.  
  
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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.
  
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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.
  
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====Comments====
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====References====
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"Full measure" is, in the Russian article, called "massive setmassive" . See also [[Singularities of differentiable mappings|Singularities of differentiable mappings]].
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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
How to Cite This Entry:
Sard theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sard_theorem&oldid=28772
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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