## Kodai Mathematical Journal

### On submanifolds with parallel mean curvature vector

#### Abstract

We consider Mn, n ≥ 3, a complete, connected submanifold of a space form $\tilde{M}^{n+p}(\tilde{c})$, 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>2n2 |H|2/(n - 1), we show that for $\tilde{c}$ ≥ 0 the codimension reduces to 1. When M is a submanifold of the unit sphere, then Mn is totally umbilic. For the case $\tilde{c}$ < 0, one imposes an additional condition that is trivially satisfied when $\tilde{c}$ ≥ 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 $\tilde{c}$ < 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

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