## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 30, Number 2 (2007), 246-251.

*L*_{n/2}-pinching theorem for submanifolds in a sphere

#### Abstract

Let *M*^{n} (*n* ≥ 2) be a n-dimensional oriented closed submanifolds with parallel mean curvature in *S*^{n + p} (1), denote by *S*, the norm square of the second fundamental form of *M*. *H* is the constant mean curvature of *M*. We prove that if ∫_{M} *S*^{n/2} ≤ *A*(*n*), where *A*(*n*) is a positive universal constant, then *M* must be a totally umbilical hypersurface in the sphere *S*^{n + 1}.

#### Article information

**Source**

Kodai Math. J., Volume 30, Number 2 (2007), 246-251.

**Dates**

First available in Project Euclid: 3 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1183475515

**Digital Object Identifier**

doi:10.2996/kmj/1183475515

**Mathematical Reviews number (MathSciNet)**

MR2343421

#### Citation

Xu, Huiqun. L n /2 -pinching theorem for submanifolds in a sphere. Kodai Math. J. 30 (2007), no. 2, 246--251. doi:10.2996/kmj/1183475515. https://projecteuclid.org/euclid.kmj/1183475515