Canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space

TL;DR

This paper introduces canonical coordinates for minimal time-like surfaces in n-dimensional Minkowski space, establishing their existence, uniqueness, and geometric properties using complex functions over double numbers.

Contribution

It develops a new framework for canonical coordinates on minimal time-like surfaces, linking them to invariants and hyperbolic geometry in Minkowski space.

Findings

Canonical coordinates are uniquely determined and expressed via surface invariants.

A complex function over double numbers characterizes these coordinates.

Geometric interpretation relates coordinates to the hyperbola of normal curvature.

Abstract

We introduce canonical coordinates on minimal time-like surfaces in the n-dimensional Minkowski space and prove the existence and the uniqueness of these parameters. With respect to these coordinates the coefficients of the first fundamental form are expressed by the invariants of the surface. On any time-like surface we introduce a special complex function over the algebra of the double numbers and apply the analysis over the algebra of the double numbers as a convenient tool to study these surfaces. Then the canonical coordinates on minimal time-like surfaces are characterized by a natural condition for this complex function. We consider the hyperbola of the normal curvature of any minimal time-like surface and give a geometric interpretation of the canonical coordinates in terms of the elements of this hyperbola.