On certain properties of pseudo-Riemannian Bertrand manifolds

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.