## Journal of the Mathematical Society of Japan

### Spherical rigidities of submanifolds in Euclidean spaces

Qing-Ming CHENG

#### 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)$S\displaystyle \leq\frac{n^{2}H^{2}}{n-1},$ or

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

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 $S^{n}(c)$, the totally geodesic Euclidean space $E^{n}$, and the generalized cylinder $S^{n-1}(c)\times E^{1}$ are only $n$-dimensional $(n>2)$ complete connected submanifolds $M^{n}$ with constant mean curvature $H$ in $E^{n+p}$ if $S\leq n^{2}H^{2}/(n-1)$ holds.

#### Article information

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

Dates
First available in Project Euclid: 3 October 2007

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

Digital Object Identifier
doi:10.2969/jmsj/1191418640

Mathematical Reviews number (MathSciNet)
MR2048469

Zentralblatt MATH identifier
1066.53013

#### 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