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Fundamental Theorems for Timelike Surfaces in the Minkowski 4-Space

Victoria Bencheva, Velichka Milousheva

Abstract

In the present paper, we study timelike surfaces free of minimal points in the four-dimensional Minkowski space. For each such surface we introduce a geometrically determined pseudo-orthonormal frame field and writing the derivative formulas with respect to this moving frame field and using the integrability conditions, we obtain a system of six functions satisfying some natural conditions. In the general case, we prove a Fundamental Bonnet-type theorem (existence and uniqueness theorem) stating that these six functions, satisfying the natural conditions, determine the surface up to a motion. In some particular cases, we reduce the number of functions and give the fundamental theorems.

Fundamental Theorems for Timelike Surfaces in the Minkowski 4-Space

Abstract

In the present paper, we study timelike surfaces free of minimal points in the four-dimensional Minkowski space. For each such surface we introduce a geometrically determined pseudo-orthonormal frame field and writing the derivative formulas with respect to this moving frame field and using the integrability conditions, we obtain a system of six functions satisfying some natural conditions. In the general case, we prove a Fundamental Bonnet-type theorem (existence and uniqueness theorem) stating that these six functions, satisfying the natural conditions, determine the surface up to a motion. In some particular cases, we reduce the number of functions and give the fundamental theorems.
Paper Structure (3 sections, 3 theorems, 51 equations)

This paper contains 3 sections, 3 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fundamental theorems

Key Result

Theorem 3.2

Let $f(u,v) > 0$, $\nu(u,v)$, $\lambda_1(u,v)$, $\mu_1(u,v)$, $\lambda_2(u,v)$, $\mu_2(u,v)$, $\mu_1 \mu_2 \neq 0$ be six smooth functions, defined in a domain ${\mathcal{D}}, \,\, {\mathcal{D}} \subset {\mathbb R}^2$, and satisfying the conditions If $\{x_0, y_0, (n_1)_0, (n_2)_0\}$ is a pseudo-orthonormal frame at a point $p_0 \in \mathbb R^4_1$, then there exists a subdomain ${\mathcal{D}}_0

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof