Hypersurfaces of nonnegative curvature in a Hilbert space

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.