Cut loci of Berger type Lorentzian structures

TL;DR

This paper investigates the geometric properties of Lorentzian structures deformed along Hopf fibration fibers, focusing on geodesic optimality, cut loci, and injectivity radius in the universal cover of Berger-type anti de-Sitter spaces.

Contribution

It provides a detailed analysis of cut loci, geodesic behavior, and injectivity radius for a class of Lorentzian manifolds deformed along Hopf fibers, extending understanding of their geometric structure.

Findings

Determined the cut time and cut locus for these Lorentzian manifolds.

Computed the injectivity radius of the deformed Lorentzian structures.

Characterized the attainable sets by admissible curves in the universal cover.

Abstract

Consider the deformation of the standard Lorentzian metric on the anti de-Sitter space along the fibers of the Hopf fibration. We study the universal covering of this Lorentzian manifold to exclude a priori presence of time-like cycles. We describe the sets attainable by admissible curves and study the question of the existence of the longest arcs. Next, we investigate Lorentzian geodesics for optimality: we find the cut time and the cut locus. As a geometric application we compute the injectivity radius of the corresponding Lorentzian manifold.