Sard's theorem: Difference between revisions
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Suppose <math>M</math> and <math>N</math> are differential manifolds, and the dimension of <math>M</math> is strictly less than the dimension of <math>N</math>. Then, if <math>f:M \to N</math> is a [[smooth map]], the image <math>f(M)</math> has measure zero as a subset of <math>N</math>. In particular, <math>f</math> cannot be surjective. | Suppose <math>M</math> and <math>N</math> are differential manifolds, and the dimension of <math>M</math> is strictly less than the dimension of <math>N</math>. Then, if <math>f:M \to N</math> is a [[smooth map]], the image <math>f(M)</math> has measure zero as a subset of <math>N</math>. In particular, <math>f</math> cannot be surjective. | ||
This also shows that [[manifold is not union of images of manifolds of smaller dimension|a differential manifold cannot be expressed as a union of the images of countably many smooth maps from differential manifolds of strictly smaller dimension]] | |||
Latest revision as of 20:07, 18 May 2008
This article gives the statement and possibly proof of a theorem that discusses regular values, critical values, regular points or critical points of a smooth map between differential manifolds
Statement
Suppose and are differential manifolds and is a smooth map between them. Then, the set of regular values of is a subset of measure zero in .
Applications
Suppose and are differential manifolds, and the dimension of is strictly less than the dimension of . Then, if is a smooth map, the image has measure zero as a subset of . In particular, cannot be surjective.
This also shows that a differential manifold cannot be expressed as a union of the images of countably many smooth maps from differential manifolds of strictly smaller dimension