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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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Named after Anthony P. Morse and Arthur Sard.
Recall the following definitions from diferential topology. The set of critical points of a smooth map is the set of points in the domain of where the tangent map is not surjective. The set of critical values of is the -image of the set of critical points of . The set of regular values of is the complement of the set of critical values of .
Suppose and are smooth manifolds of dimension and respectively and is a -smooth map, where and . Then the set of critical values in is a meager subset (alias first category subset) and a negligible subset (alias measure zero subset) of . In particular, the set of regular values is dense in . Furthermore, the -image of points of where has rank at most () has Hausdorff dimension at most .
If is a Banach manifold and , is a Fredholm map, and is strictly greater than the index of , then the critical values of form a meager subset of .
The case :
The case :
The case when is a Banach manifold:
The part concerning Hausdorff measures:
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