Biconservative surfaces in the 4-dimensional Euclidean sphere

Abstract

In this paper, we study biconservative surfaces with parallel normalized mean curvature vector field (PN MC) in the 4-dimensional unit Euclidean sphere 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) PN MC biconservative immersion in S-4. Then, we obtain the local parametrization of these surfaces in the 5-dimensional Euclidean space E-5. We end the paper by proving that the substantial codimension of PN MC biconservative surfaces in S-n, n = 5, is equal to 2.