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

Abstract

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.

General Rotational Surfaces in Pseudo-Euclidean 4-Space with Neutral Metric

“…Remark 3.3. In [2], an invariant local theory of Lorentz surfaces in the pseudo-Euclidean space E 4 2 is developed and a family of eight geometric functions is introduced. It is proved that these geometric functions determine the surface up to a rigid motion in E 4 2 .…”

“…It is proved that these geometric functions determine the surface up to a rigid motion in E 4 2 . The functions ν 1 , ν 2 , µ, γ 2 , β 2 given in (9) are the geometric functions of the general rotational surface M 1 in the sense of [2] (note that the other three geometric functions of M 1 are equal to zero).…”

“…These five functions are the geometric functions of the general rotational surface of hyperbolic type M 2 in the sense of [2]. The other three geometric functions of M 2 are equal to zero.…”

“…Let E 4 2 be the pseudo-Euclidean 4-space endowed with the canonical pseudo-Euclidean metric of index 2 given by g 0 = dx 2 1 + dx 2 2 − dx 2 3 − dx 2 4 , where (x 1 , x 2 , x 3 , x 4 ) is a rectangular coordinate system of E 4 2 . As usual, we denote by .…”

Surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space

“…In [1] we studied the local theory of Lorentz surfaces with parallel normalized mean curvature vector field in the pseudo-Euclidean space with neutral metric E 4 2 . Introducing special geometric parameters (called canonical parameters) on each such surface, we described the Lorentz surfaces with parallel normalized mean curvature vector field in terms of three invariant functions satisfying a system of three partial differential equations.…”