## Kodai Mathematical Journal

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

#### Abstract

Let M be an n-dimensional closed submanifold with parallel mean curvature in Sn+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

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