## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 22, Number 5 (2015), 707-713.

### On the geometry of complete submanifolds immersed in the hyperbolic space

Henrique F. de Lima, Fábio R. dos Santos, and Marco Antonio L. Velásquez

#### Abstract

We deal with $n$-dimensional complete submanifolds immersed with parallel nonzero mean curvature vector ${\bf H}$ in the hyperbolic space $\mathbb{H}^{n+p}$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $M^n$ must be pseudo-umbilical, which means that ${\bf H}$ is an umbilical direction. In particular, we conclude that $M^n$ is a minimal submanifold of a small hypersphere of $\mathbb{H}^{n+p}$.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 5 (2015), 707-713.

**Dates**

First available in Project Euclid: 17 December 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1450389242

**Digital Object Identifier**

doi:10.36045/bbms/1450389242

**Mathematical Reviews number (MathSciNet)**

MR3435076

**Zentralblatt MATH identifier**

1332.53081

**Subjects**

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Secondary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 53C50: Lorentz manifolds, manifolds with indefinite metrics

**Keywords**

Hyperbolic space complete submanifolds parallel mean curvature vector pseudo-umbilical submanifolds minimal submanifolds

#### Citation

de Lima, Henrique F.; dos Santos, Fábio R.; Velásquez, Marco Antonio L. On the geometry of complete submanifolds immersed in the hyperbolic space. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 5, 707--713. doi:10.36045/bbms/1450389242. https://projecteuclid.org/euclid.bbms/1450389242