Hypersurfaces of nonnegative curvature in a Hilbert space

Abstract

In this paper we prove the following generalizations of known theorems about hypersurfaces in R n {{\mathbf {R}}^n} : Let M M be a hypersurface in a Hilbert space. (1) If on M M the sectional curvature K ( σ ) K(\sigma ) is nonnegative for every 2 2 -plane section σ \sigma and if K ( σ ) > 0 K(\sigma ) > 0 for at least one σ \sigma , then M M is the boundary of a convex body. (2) If K ( σ ) = 0 K(\sigma ) = 0 for all σ \sigma , then M M is a hypercylinder. The main tool in these theorems is Smale’s infinite dimensional Sard’s theorem.