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

## TL;DR

This paper introduces a new class of Lorentz surfaces called meridian surfaces in pseudo-Euclidean 4-space with a lightlike axis, classifies those with constant Gauss curvature, and explores their mean curvature properties.

## Contribution

It provides a complete classification of meridian surfaces with constant Gauss curvature and analyzes their mean curvature vector fields in the context of pseudo-Euclidean space.

## Key findings

- Classified meridian surfaces with constant Gauss curvature.
- Proved no meridian surfaces have parallel mean curvature vector field except CMC surfaces in a hyperplane.
- Identified meridian surfaces with parallel normalized mean curvature vector but not parallel mean curvature vector.

## Abstract

We construct a special class of Lorentz surfaces in the pseudo-Euclidean 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.