On the Theory of Lorentz Surfaces with Parallel Normalized Mean Curvature Vector Field in Pseudo-Euclidean 4-Space

TL;DR

This paper develops a local invariant theory for Lorentz surfaces in pseudo-Euclidean 4-space, characterizing surfaces with parallel normalized mean curvature vector fields using minimal invariant functions and PDEs.

Contribution

It introduces a new invariant framework and canonical parameters to classify Lorentz surfaces with parallel normalized mean curvature vectors, solving a specific geometric PDE problem.

Findings

Existence and uniqueness theorem for the invariant functions

Canonical parameters uniquely determine the surface up to rigid motion

Reduction of the classification problem to three PDEs

Abstract

We develop an invariant local theory of Lorentz surfaces in pseudo-Euclidean 4-space by use of a linear map of Weingarten type. We find a geometrically determined moving frame field at each point of the surface and obtain a system of geometric functions. We prove a fundamental existence and uniqueness theorem in terms of these functions. On any Lorentz surface with parallel normalized mean curvature vector field we introduce special geometric (canonical) parameters and prove that any such surface is determined up to a rigid motion by three invariant functions satisfying three natural partial differential equations. In this way we minimize the number of functions and the number of partial differential equations determining the surface, which solves the Lund-Regge problem for this class of surfaces.