Biconservative surfaces in the $4$-dimensional Euclidean sphere

Abstract

In this paper, we study biconservative surfaces with parallel normalized mean curvature vector field ($PNMC$) in the $4$-dimensional unit Euclidean sphere $\mathbb{S}^4$. First, we study the existence and uniqueness of such surfaces. We obtain that there exists a $2$-parameter family of non-isometric abstract surfaces that admit a (unique) $PNMC$ biconservative immersion in $\mathbb{S}^4$. Then, we obtain the local parametrization of these surfaces in the $5$-dimensional Euclidean space $\mathbb{E}^5$.