## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 32, Number 1 (2009), 59-76.

### On submanifolds with parallel mean curvature vector

Kellcio O. Araújo and Keti Tenenblat

#### Abstract

We consider *M*^{n}, *n* ≥ 3, a complete, connected submanifold of a space form , whose non vanishing mean curvature vector *H* is parallel in the normal bundle. Assuming the second fundamental form *h* of *M* satisfies the inequality <*h*>^{2} ≤ *n*^{2} |*H*|^{2}/(*n* - 1), we show that for ≥ 0 the codimension reduces to 1. When *M* is a submanifold of the unit sphere, then *M*^{n} is totally umbilic. For the case < 0, one imposes an additional condition that is trivially satisfied when ≥ 0. When *M* is compact and has non-negative Ricci curvature then it is a geodesic hypersphere in the hyperbolic space. An alternative additional condition, when < 0, reduces the codimension to 3.

#### Article information

**Source**

Kodai Math. J., Volume 32, Number 1 (2009), 59-76.

**Dates**

First available in Project Euclid: 1 April 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1238594546

**Digital Object Identifier**

doi:10.2996/kmj/1238594546

**Mathematical Reviews number (MathSciNet)**

MR2518554

**Zentralblatt MATH identifier**

1160.53026

#### Citation

Araújo, Kellcio O.; Tenenblat, Keti. On submanifolds with parallel mean curvature vector. Kodai Math. J. 32 (2009), no. 1, 59--76. doi:10.2996/kmj/1238594546. https://projecteuclid.org/euclid.kmj/1238594546