On certain properties of pseudo-Riemannian Bertrand manifolds

TL;DR

This paper investigates properties of pseudo-Riemannian Bertrand manifolds, establishing that no completely Bertrand systems exist in pseudo-Riemannian 2D revolution manifolds, and discusses future research directions.

Contribution

It proves the non-existence of pseudo-Riemannian completely Bertrand systems and provides an overview of the current theory and open problems.

Findings

No pseudo-Riemannian completely Bertrand systems exist.

Provides an overview of pseudo-Riemannian Bertrand manifold properties.

Discusses future research problems.

Abstract

In the present article we prove and discuss several properties of pseudo-Riemannian Bertrand manifolds, and give an overview of the state of the theory together with problems for future work. In particular, we prove that there are no pseudo-Riemannian completely Bertrand systems -- dynamical systems of movement in a central field on a pseudo-Riemannian 2-dimensional manifold of revolution such that all trajectories of movement are closed.