Sard theorem
This category corresponds roughly to MSC {{{id}}} {{{title}}}; see {{{id}}} at MathSciNet and {{{id}}} at zbMATH.
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 . 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
| [1] | A. Sard, "The measure of critical values of differentiable maps" Bull. Amer. Math. Soc. , 48 (1942) pp. 883–890 MR7523 Template:ZBL |
 |  |
