Lorentz Surfaces Pseudo-Euclidean Space Canonical Coordinates Natural Equations
Issue Date:
3-Dec-2021
Publisher:
MDPI
Citation:
Kanchev, K.; Kassabov, O.; Milousheva, V. Canonical Coordinates and Natural Equation for Lorentz Surfaces in R^3_1. Mathematics, 2021, 9, 3121. https://doi.org/ 10.3390/math9233121
Series/Report no.:
Mathematics;9, 3121
Abstract:
We consider Lorentz surfaces in R^3_1 satisfying the condition H^2−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.
