Curves in a spacelike hypersurface in the Minkowski space-time

TL;DR

This paper explores the geometry of curves in spacelike hypersurfaces within Minkowski space, focusing on hyperbolic and de Sitter surfaces, their singularities, and geometric interpretations, advancing understanding of extrinsic curve geometry in relativity contexts.

Contribution

It introduces a new analysis of hyperbolic and de Sitter surfaces associated with curves in Minkowski space using singularity theory, providing insights into their shapes and geometric meanings.

Findings

Characterization of hyperbolic and de Sitter surfaces of curves

Description of generic singularities of these surfaces

Insights into the extrinsic geometry of curves in Minkowski space

Abstract

Submanifolds in Lorentz-Minkowski space are investigated from various mathematical viewpoints and are of interest also in relativity theory. We define the hyperbolic surface and the de Sitter surface of a curve in the spacelike hypersurface M in the Minkowski 4-space. These surfaces are respectively located in the hyperbolic 3-space and in the de Sitter 3-space. We use techniques of the theory of singularities in order to describe the generic shape of these surfaces and of their singular value sets. We also investigate geometric meanings of those singularities. The results in this paper contribute to the study of the extrinsic geometry of curves in different ambient spaces.