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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Surjective mappings whose differential is nowhere surjective


Author: Y. Yomdin
Journal: Proc. Amer. Math. Soc. 111 (1991), 267-270
MSC: Primary 58C25
MathSciNet review: 1039267
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Abstract: Examples of $ {C^k}$-mappings $ f:{\mathbb{R}^n} \to {\mathbb{R}^m},n \geq m > 2$, are given, with rank $ df(x) \leq s$ at any $ x \in {\mathbb{R}^n},2 \leq s < m$, but $ f({\mathbb{R}^n}) = {\mathbb{R}^m}$, for any $ k < (n - s + 2)/(m - s + 2)$. Thus a weak form of Sard's theorem (if all the points in the source are critical, the image has measure zero) does not hold for mappings of low smoothness.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1039267-2
PII: S 0002-9939(1991)1039267-2
Article copyright: © Copyright 1991 American Mathematical Society