Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cut loci of Berger type Lorentzian structures

A. V. Podobryaev

TL;DR

The paper studies Lorentzian Berger-type structures on the universal cover of $\mathbb{H}^{1,n}$, focusing on longest Lorentzian arcs, geodesic optimality, and the cut locus. It develops a 3D invariant control framework on $G=\widetilde{\mathrm{U}}_{1,1}$, applies Pontryagin maximum principle, and analyzes three regimes determined by $\eta=\frac{I_2}{I_1}-1$: oblate, symmetric, and prolate. The authors obtain explicit attainable sets, characterize geodesics, determine conjugate and Maxwell times, and identify the cut locus and injectivity radius, with a general reduction to the higher-dimensional case via $\widetilde{U}_n$-symmetry. The results provide complete Lorentzian geometric data (distances, cut loci, injectivity radii) for Berger-type Lorentzian manifolds, enabling precise understanding of optimal trajectories and global geometry on these homogeneous spaces.

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.

Cut loci of Berger type Lorentzian structures

TL;DR

The paper studies Lorentzian Berger-type structures on the universal cover of , focusing on longest Lorentzian arcs, geodesic optimality, and the cut locus. It develops a 3D invariant control framework on , applies Pontryagin maximum principle, and analyzes three regimes determined by : oblate, symmetric, and prolate. The authors obtain explicit attainable sets, characterize geodesics, determine conjugate and Maxwell times, and identify the cut locus and injectivity radius, with a general reduction to the higher-dimensional case via -symmetry. The results provide complete Lorentzian geometric data (distances, cut loci, injectivity radii) for Berger-type Lorentzian manifolds, enabling precise understanding of optimal trajectories and global geometry on these homogeneous spaces.

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.
Paper Structure (12 sections, 39 theorems, 116 equations, 5 figures)

This paper contains 12 sections, 39 theorems, 116 equations, 5 figures.

Table of Contents

  1. Optimal control problem in the three dimensional case
  2. Lorentzian geodesics
  3. Attainable sets and existence of the solution
  4. The oblate case
  5. The symmetric case
  6. The prolate case
  7. Optimality of geodesics
  8. The first conjugate time
  9. The first Maxwell time for symmetries
  10. The cut locus
  11. The injectivity radius
  12. General case

Key Result

Theorem A

We have the following description of the attainable set $\mathcal{A}$. (1) In the oblate case we can attain any point of the manifold $M$, i.e., $\mathcal{A} = M$. The longest arcs do not exist. (2) In the symmetric case The longest arcs exist only for the terminal points located in the set (3) In the prolate case where $\eta = \frac{I_2}{I_1} - 1$. Any point of the set $\mathcal{A}$ can be rea

Figures (5)

  • Figure 1: The attainable sets for the symmetric case (the surface of revolution with respect to the $c$-axis). The solid line is $c = \mathop{\mathrm{\mathrm{arctan}}}\nolimits{|w|}$, the dashed lines are $c = \pi k \pm \mathop{\mathrm{\mathrm{arctan}}}\nolimits{|w|}$, $k \in \mathbb{N}$. The dashed lines do not belong to the sets. Note that in spite of $c$ taking the first position in the pair $(c,w)$ the $c$-axis is vertical.
  • Figure 2: The sequence of arcs with the Lorentzian length tending to infinity.
  • Figure 3: The attainable set for the prolate case where $\eta = 0.1$ (the surface of revolution with respect to the $c$-axis).
  • Figure 4: Geodesics that lose optimality (the solid lines), geodesics that are optimal to infinity (the dashed lines) and the Maxwell points (the thick points on the $c$-axis). The surface of revolution with respect to $c$-axis.
  • Figure 5: The wavefronts (the solid lines) and the Maxwell points (the thick points on the $c$-axis). The dashed lines are the wavefront of zero radius (the light-like geodesics). The surface of revolution with respect to $c$-axis.

Theorems & Definitions (83)

  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 1: finsler
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1: Pontryagin's maximum principle pontryaginagrachev-sachkov
  • Definition 1
  • ...and 73 more