Conformal geometry of timelike curves in the (1+2)-Einstein universe

TL;DR

This paper investigates the conformal geometry of timelike curves in the (1+2)-Einstein universe, focusing on invariants, existence of closed trajectories, and connections to knot theory in a compactified Minkowski space.

Contribution

It introduces new conformal invariants for timelike curves and explores conditions for closed trajectories within the Einstein universe.

Findings

Identification of local and global conformal invariants.

Existence criteria for closed conformal trajectories.

Connections established between conformal geometry and knot theory.

Abstract

We study the conformal geometry of timelike curves in the (1+2)-Einstein universe, the conformal compactification of Minkowski 3-space defined as the quotient of the null cone of $\mathbb{R}^{2,3}$ by the action by positive scalar multiplications. The purpose is to describe local and global conformal invariants of timelike curves and to address the question of existence and properties of closed trajectories for the conformal strain functional. Some relations between the conformal geometry of timelike curves and the geometry of knots and links in the 3-sphere are discussed.