BONNET SURFACES IN FOUR-DIMENSIONAL SPACE FORMS

Abstract

We study isometric deformations of surfaces in four-dimensional space forms preserving the length of the mean curvature vector. In particular we consider the natural condition, called to be simple, and show that such surfaces with flat normal bundle are Bonnet surfaces in totally geodesic or umbilic 3-dimensional space forms, which is regarded as a generalization of Chen–Yau's reduction theorem for surfaces with parallel mean curvature vector.