Convexity of Hypersurfaces in Spherical Spaces

TL;DR

This paper proves that complete locally-convex immersions of (n-1)-manifolds into n-spheres are surjective onto convex set boundaries for dimensions n >= 3, advancing understanding of convexity in spherical geometry.

Contribution

It establishes a key property of locally-convex immersions in spherical spaces, showing they cover the boundary of convex sets for dimensions n >= 3.

Findings

Complete locally-convex immersions are surjective onto convex set boundaries for n >= 3

The result extends convexity theory to spherical hypersurfaces

Provides a characterization of convex hypersurfaces in spherical spaces

Abstract

A spherical set is called convex if for every pair of its points there is at least one minimal geodesic segment that joins these points and lies in the set. We prove that for n >= 3 a complete locally-convex (topological) immersion of a connected (n-1)-manifold into the n-sphere is a surjection onto the boundary of a convex set.