Rigidity of the timelike marked length spectrum and length-twist coordinates of singular de-Sitter tori

TL;DR

This paper investigates the uniqueness and rigidity of closed timelike geodesics in singular de-Sitter tori, introducing length spectrum and length-twist coordinates to understand their geometric structure.

Contribution

It establishes the rigidity of the timelike marked length spectrum and constructs length-twist coordinates for singular de-Sitter tori, advancing understanding of their geometric properties.

Findings

Uniqueness of closed timelike geodesics in their homotopy class.

Rigidity of the timelike marked length spectrum.

Construction of length-twist coordinates for the deformation space.

Abstract

In this paper, we study the closed timelike geodesics of de-Sitter tori with one singularity and prove their uniqueness in their free homotopy class. We introduce the notion of timelike marked length spectrum of such a torus, and establish its rigidity with respect to the lengths of two homotopy classes of intersection number one. We also construct length-twist coordinates on the deformation space of de-Sitter tori with one singularity.