## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 34, Number 1 (2011), 85-104.

### An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere

Hong-Wei Xu, Fei Huang, and Fei Xiang

#### Abstract

Let *M* be an *n*-dimensional closed submanifold with parallel mean curvature in *S*^{n+p}, the trace free part of the second fundamental form, and (*u*) = ||(*u*, *u*)||^{2} for any unit vector *u* *TM*. We prove that there exists a positive constant *C*(*n*, *p*, *H*) (≥ 1/3) such that if (*u*) ≤ *C*(*n*, *p*, *H*), then either (*u*) ≡ 0 and *M* is a totally umbilical sphere, or (*u*) ≡ *C*(*n*, *p*, *H*). A geometrical classification of closed submanifolds with parallel mean curvature satisfying (*u*) ≡ *C*(*n*, *p*, *H*) is also given. Our main result is an extension of the Gauchman theorem [4].

#### Article information

**Source**

Kodai Math. J., Volume 34, Number 1 (2011), 85-104.

**Dates**

First available in Project Euclid: 31 March 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1301576764

**Digital Object Identifier**

doi:10.2996/kmj/1301576764

**Mathematical Reviews number (MathSciNet)**

MR2786783

**Zentralblatt MATH identifier**

1220.53068

#### Citation

Xu, Hong-Wei; Huang, Fei; Xiang, Fei. An extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere. Kodai Math. J. 34 (2011), no. 1, 85--104. doi:10.2996/kmj/1301576764. https://projecteuclid.org/euclid.kmj/1301576764