## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 56, Number 2 (2004), 475-487.

### Spherical rigidities of submanifolds in Euclidean spaces

#### Abstract

In this paper, we study $n$-dimensional complete immersed submanifolds in a Euclidean space ${E}^{n+p}$. We prove that if ${M}^{n}$ is an $n$-dimensional compact connected immersed submanifold with nonzero mean curvature $H$ in ${E}^{n+p}$ and satisfies either:

(1)$S\le \frac{{n}^{2}{H}^{2}}{n-1},$ or

(2)${n}^{2}{H}^{2}\le \frac{(n-1)R}{n-2}$,

then ${M}^{n}$ is diffeomorphic to a standard $n$-sphere, where $S$ and $R$ denote the squared norm of the second fundamental form of ${M}^{n}$ and the scalar curvature of ${M}^{n}$, respectively.

On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere ${S}^{n}\left(c\right)$, the totally geodesic Euclidean space ${E}^{n}$, and the generalized cylinder ${S}^{n-1}\left(c\right)\times {E}^{1}$ are only $n$-dimensional $(n>2)$ complete connected submanifolds ${M}^{n}$ with constant mean curvature $H$ in ${E}^{n+p}$ if $S\le {n}^{2}{H}^{2}/(n-1)$ holds.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 56, Number 2 (2004), 475-487.

**Dates**

First available in Project Euclid: 3 October 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1191418640

**Digital Object Identifier**

doi:10.2969/jmsj/1191418640

**Mathematical Reviews number (MathSciNet)**

MR2048469

**Zentralblatt MATH identifier**

1066.53013

**Subjects**

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

**Keywords**

Submanifolds differentiable sphere locally convex hypersurfaces generalized cylinder mean curvature squared norm of the second fundamental form

#### Citation

CHENG, Qing-Ming. Spherical rigidities of submanifolds in Euclidean spaces. J. Math. Soc. Japan 56 (2004), no. 2, 475--487. doi:10.2969/jmsj/1191418640. https://projecteuclid.org/euclid.jmsj/1191418640