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July, 2013 A simple improvement of a differentiable classification result for complete submanifolds
Ezequiel R. BARBOSA
J. Math. Soc. Japan 65(3): 787-796 (July, 2013). DOI: 10.2969/jmsj/06530787

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 $\mathbb{R}^n$.

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Ezequiel R. BARBOSA. "A simple improvement of a differentiable classification result for complete submanifolds." J. Math. Soc. Japan 65 (3) 787 - 796, July, 2013. https://doi.org/10.2969/jmsj/06530787

Information

Published: July, 2013
First available in Project Euclid: 23 July 2013

zbMATH: 1275.53049
MathSciNet: MR3084981
Digital Object Identifier: 10.2969/jmsj/06530787

Subjects:
Primary: 53C40
Secondary: 53C20 , 53C44

Keywords: differentiable sphere theorem , Ricci flow , second fundamental form , sectional curvature , submanifold

Rights: Copyright © 2013 Mathematical Society of Japan

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Vol.65 • No. 3 • July, 2013
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