Canonical Coordinates and Natural Equations for Minimal Time-like Surfaces in R⁴₂

2019/11/29 by Ganchev, Georgi, Kanchev, Krasimir
#53A10 (Primary) #53B30 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.00014

Abstract

We apply the complex analysis over the double numbers D to study the minimal time-like surfaces in R⁴₂. A minimal time-like surface which is free of degenerate points is said to be of general type. We divide the minimal time-like surfaces of general type into three types and prove that these surfaces admit special geometric (canonical) parameters. Then the geometry of the minimal time-like surfaces of general type is determined by the Gauss curvature K and the curvature of the normal connection \varkappa, satisfying the system of natural equations for these surfaces. We prove the following: If (K, \varkappa), K²- \varkappa² > 0 is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the first type and exactly one minimal time-like surface of the second type with invariants (K, \varkappa); if (K, \varkappa), K²- \varkappa² < 0 is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the third type with invariants (K, \varkappa).

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