Biconservative Submanifolds in $$\mathbb {S}^n\times \mathbb {R}$$ S n × R and $$\mathbb {H}^n\times \mathbb {R}$$ H n × 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 codimesion 2. We obtain some sufficient and necessary 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 \(\mathbb {H}^n\times \mathbb {R}\).