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 \(\mathbb {E}^5\). We end the paper by proving that the substantial codimension of PNMC biconservative surfaces in \(\mathbb {S}^n\), \(n\ge 5\), is equal to 2.