Journal of the Mathematical Society of Japan

A simple improvement of a differentiable classification result for complete submanifolds

Ezequiel R. BARBOSA

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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 $\mathbb{R}^n$.

Article information

J. Math. Soc. Japan, Volume 65, Number 3 (2013), 787-796.

First available in Project Euclid: 23 July 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.)

submanifold second fundamental form differentiable sphere theorem Ricci flow sectional curvature


BARBOSA, Ezequiel R. A simple improvement of a differentiable classification result for complete submanifolds. J. Math. Soc. Japan 65 (2013), no. 3, 787--796. doi:10.2969/jmsj/06530787.

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