An Invariant Theory of Spacelike Surfaces in the Four-dimensional Minkowski Space

TL;DR

This paper develops an invariant geometric theory for spacelike surfaces in four-dimensional Minkowski space, introducing new invariants and a fundamental theorem that characterizes these surfaces up to motion.

Contribution

It introduces a new invariant linear map, principal lines, and a moving frame for spacelike surfaces, leading to a fundamental theorem and classification based on invariants.

Findings

Eight invariant functions characterize the surface.

Basic classes are described via tangent indicatrix and normal curvature ellipse.

Application to spacelike general rotational surfaces.

Abstract

We consider spacelike surfaces in the four-dimensional Minkowski space and introduce geometrically an invariant linear map of Weingarten-type in the tangent plane at any point of the surface under consideration. This allows us to introduce principal lines and an invariant moving frame field. Writing derivative formulas of Frenet-type for this frame field, we obtain eight invariant functions. We prove a fundamental theorem of Bonnet-type, stating that these eight invariants under some natural conditions determine the surface up to a motion. We show that the basic geometric classes of spacelike surfaces in the four-dimensional Minkowski space, determined by conditions on their invariants, can be interpreted in terms of the properties of the two geometric figures: the tangent indicatrix, and the normal curvature ellipse. We apply our theory to a class of spacelike general rotational surfaces.