Canonical Coordinates and Natural Equations for Minimal Time-like Surfaces in $R_2^4$

Abstract

We apply the complex analysis over the double numbers $D$ to study the minimal time-like surfaces in $R_2^4$. 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 $(K,\varkappa ),K^2-{\varkappa }^2>0$ 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 $(K,\varkappa ),K^2-{\varkappa }^2<0$ is a solution to the system of natural equations, then there exists exactly one minimal time-like surface of the third type with invariants $(K,\varkappa )$.

arXiv comment52 pages. arXiv admin note: text overlap with arXiv:1911.10779

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Cited by in corpus (1)

• Minimal Timelike Surfaces in the Lorentz-Minkowski 3-space and Their Canonical Parameters