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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Hypersurfaces of nonnegative curvature in a Hilbert space


Author: Leo Jonker
Journal: Trans. Amer. Math. Soc. 169 (1972), 461-474
MSC: Primary 58B20
DOI: https://doi.org/10.1090/S0002-9947-1972-0307271-8
MathSciNet review: 0307271
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Abstract: In this paper we prove the following generalizations of known theorems about hypersurfaces in ${{\mathbf {R}}^n}$: Let $M$ be a hypersurface in a Hilbert space. (1) If on $M$ the sectional curvature $K(\sigma )$ is nonnegative for every $2$-plane section $\sigma$ and if $K(\sigma ) > 0$ for at least one $\sigma$, then $M$ is the boundary of a convex body. (2) If $K(\sigma ) = 0$ for all $\sigma$, then $M$ is a hypercylinder. The main tool in these theorems is Smale’s infinite dimensional Sard’s theorem.


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Keywords: Hilbert manifold, hypersurface, positive curvature, convex, Sard’s theorem, transverse, cylinder, Fredholm map
Article copyright: © Copyright 1972 American Mathematical Society