Biconservative submanifolds in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$

Abstract

In this paper, we study biconservative submanifolds in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ with parallel mean curvature vector field and co-dimension 2. We obtain some necessary and sufficient conditions for such submanifolds to be conservative. In particular, we obtain a complete classification of 3-dimensional biconservative submanifolds in ${\mathbb{S}}^4\times \mathbb{R}$ and ${\mathbb{H}}^4\times \mathbb{R}$ with nonzero parallel mean curvature vector field. We also get some results for biharmonic submanifolds in ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$.

arXiv comment17 pages