Topological and geometric rigidity of nonnegatively curved submanifolds

TL;DR

This paper explores how specific curvature bounds influence the topology and geometry of compact submanifolds in nonnegatively curved space forms, revealing strong constraints and sharpness of results.

Contribution

It establishes new rigidity results linking sectional curvature bounds to topological and geometric properties of submanifolds in nonnegative curvature spaces.

Findings

Curvature bounds impose strong topological constraints.

Examples demonstrate the sharpness of the theoretical results.

Results unify geometric and topological rigidity phenomena.

Abstract

We investigate the topology and geometry of compact submanifolds in space forms of nonnegative curvature that satisfy a lower bound on the sectional curvature, depending only on the length of the mean curvature vector of the immersion. We show that this condition imposes strong constraints on either the topology or geometry of the submanifold. Additionally, we provide examples that demonstrate the sharpness of our result.