A simple improvement of a differentiable classification result for complete submanifolds

Abstract

We consider $M^n$, $n\geq3$, an $n$-dimensional complete submanifold of a Riemannian manifold $(\overline{M}^{n+p},\overline{g})$. We prove that if for all point $x\in M^n$ the following inequality is satisfied

$$S\leq\frac{8}{3} \bigg( \overline{K}_{\min}-\frac{1}{4}\overline{K}_{\max} \bigg)+\frac{n^2H^2}{n-1},$$

with strictly inequality at one point, where $S$ and $H$ denote the squared norm of the second fundamental form and the mean curvature of $M^n$ respectively, then $M^n$ is either diffeomorphic to a spherical space form or the Euclidean space $\mathbb{R}^n$. In particular, if $M^n$ is simply connected, then $M^n$ is either diffeomorphic to the sphere $\mathbb{S}^n$ or the Euclidean space $\mathbb{R}^n$.