Canonical Coordinates and Natural Equation for Lorentz Surfaces in R13

Krasimir Kanchev,Ognian Kassabov,Velichka Milousheva

doi:10.3390/math9233121

Abstract

We consider Lorentz surfaces in R13 satisfying the condition H2−K≠0, where K and H are the Gaussian curvature and the mean curvature, respectively, and call them Lorentz surfaces of general type. For this class of surfaces, we introduce special isotropic coordinates, which we call canonical, and show that the coefficient F of the first fundamental form and the mean curvature H, expressed in terms of the canonical coordinates, satisfy a special integro-differential equation which we call a natural equation of the Lorentz surfaces of a general type. Using this natural equation, we prove a fundamental theorem of Bonnet type for Lorentz surfaces of a general type. We consider the special cases of Lorentz surfaces of constant non-zero mean curvature and minimal Lorentz surfaces. Finally, we give examples of Lorentz surfaces illustrating the developed theory.

Highlights

The question of describing surfaces with a prescribed mean or Gaussian curvature in the Euclidean 3-space R3 and in the other Riemannian space forms have been the subject of an intensive study
The geometry of spacelike or timelike surfaces in the Minkowski 3-space R31 has been of wide interest
A Kenmotsu-type representation formula for spacelike surfaces with prescribed mean curvature was obtained by Akutagawa and Nishikawa in [1]

Summary

Introduction

The question of describing surfaces with a prescribed mean or Gaussian curvature in the Euclidean 3-space R3 and in the other Riemannian space forms have been the subject of an intensive study. Timelike surfaces in R31 with prescribed Gaussian curvature and Gauss map are studied in [4], where a Kenmotsu-type representation for such surfaces is given. Canonical principal coordinates are introduced for this class of surfaces, and a natural nonlinear partial differential equation is derived, which is equivalent to (3) in the case of a minimal surface. It can be reduced to (3) by changing the isotropic coordinates with isothermal ones This shows that the newly obtained results for an arbitrary Lorentz surface of a general type in R31 generalize the known results for the case of a minimal Lorentz surface.

Preliminaries

Canonical Isotropic Coordinates of Lorentz Surfaces in R31

Natural Equation of Lorentz Surfaces of General Type in R31

Examples