Topological and differentiable rigidity of submanifolds in space forms

TL;DR

This paper establishes topological and differentiable sphere theorems for compact submanifolds in space forms under Ricci curvature pinching conditions, demonstrating that certain curvature bounds imply the submanifold is a sphere.

Contribution

It proves a sharp Ricci curvature pinching condition ensuring a submanifold is homeomorphic to a sphere and introduces a new differentiable sphere theorem for positively Ricci curved submanifolds.

Findings

Submanifolds with Ricci curvature exceeding a specific bound are homeomorphic to spheres.

The curvature pinching condition provided is sharp.

A new differentiable sphere theorem for submanifolds with positive Ricci curvature is established.

Abstract

Let $F^{n+p}(c)$ be an $(n+p)$-dimensional simply connected space form with nonnegative constant curvature $c$. We prove that if $M^n(n\geq4)$ is a compact submanifold in $F^{n+p}(c)$, and if $Ric_M>(n-2)(c+H^2),$ where $H$ is the mean curvature of $M$, then $M$ is homeomorphic to a sphere. We also show that the pinching condition above is sharp. Moreover, we obtain a new differentiable sphere theorem for submanifolds with positive Ricci curvature.