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

## TL;DR

This paper classifies 3-dimensional biconservative submanifolds with parallel mean curvature in certain product spaces and explores related biharmonic submanifolds, advancing understanding of their geometric properties.

## Contribution

It provides a complete classification of specific biconservative submanifolds in product spaces and establishes conditions for their conservativity.

## Key findings

- Complete classification of 3D biconservative submanifolds in $ ext{S}^4\times \mathbb{R}$ and $ ext{H}^4\times \mathbb{R}$
- Necessary and sufficient conditions for conservativity of these submanifolds
- Results on biharmonic submanifolds in the same ambient spaces

## 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 $\mathbb{H}^{n}\times \mathbb{R}$.