# The topological rigidity theorem for submanifolds in space forms

## TL;DR

This paper proves a topological sphere theorem for compact submanifolds in space forms under optimal curvature pinching conditions, extending previous results and establishing new topological sphere criteria.

## Contribution

It establishes a new topological sphere theorem for submanifolds with optimal curvature pinching conditions, improving previous results and providing new criteria for topological sphere theorems.

## Key findings

- Submanifolds with scalar curvature exceeding a specific bound are homeomorphic to spheres.
- The curvature pinching conditions used are proven to be optimal.
- New topological sphere theorems are derived for submanifolds with pinched scalar and Ricci curvatures.

## Abstract

Let $M$ be an $n(\geq 4)$-dimensional compact submanifold in the simply connected space form $F^{n+p}(c)$ with constant curvature $c\geq 0$, where $H$ is the mean curvature of $M$. We verify that if the scalar curvature of $M$ satisfies $R>n(n-2)(c+H^2)$, and if $Ric_M\geq (n-2-\frac{2\sigma_n}{2n-\sigma_n})(c+H^2)$, then $M$ is homeomorphic to a sphere. Here $\sigma_n=sgn(n-4)((-1)^n+3)$, and $sgn(\cdot)$ is the standard sign function. This improves our previous sphere theorem \cite{XG2}. It should be emphasized that our pinching conditions above are optimal. We also obtain some new topological sphere theorems for submanifolds with pinched scalar curvature and Ricci curvature.