Meridian Surfaces on Rotational Hypersurfaces with Lightlike Axis in ${\mathbb E}^4_2$

Abstract

Abstract. We construct a special class of Lorentz surfaces in the pseudoEuclidean 4-space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with lightlike axis and call them meridian surfaces. We give the complete classification of the meridian surfaces with constant Gauss curvature and prove that there are no meridian surfaces with parallel mean curvature vector field other than CMC surfaces lying in a hyperplane. We also classify the meridian surfaces with parallel normalized mean curvature vector field. We show that in the family of the meridian surfaces there exist Lorentz surfaces which have parallel normalized mean curvature vector field but not parallel mean curvature vector. Keywords. Meridian surfaces · constant Gauss curvature · parallel mean curvature vector · parallel normalized mean curvature vector. MSC 2010 Classification. Primary: 53A35; Secondary:53B30 · 53B25.