An optimal lower curvature bound for convex hypersurfaces in Riemannian manifolds

TL;DR

This paper proves that convex hypersurfaces in Riemannian manifolds with sectional curvature at least k are Alexandrov spaces with curvature at least k, establishing an optimal lower curvature bound for such hypersurfaces.

Contribution

It provides the first proof that convex hypersurfaces in manifolds with curvature ≥ k are Alexandrov spaces with curvature ≥ k, refining Buyalo's earlier theorem.

Findings

Convex hypersurfaces inherit a curvature bound of at least k.

The theorem is optimal, matching known bounds.

Extends classical results to a broader class of hypersurfaces.

Abstract

It is proved that a convex hypersurface in a Riemannian manifold of sectional curvature at least k is an Alexandrov's space of curvature at least k. This theorem provides an optimal lower curvature bound for an older theorem of Buyalo.