On geodesic mappings of symmetric pairs

Abstract

The paper treats properties of pseudo-Riemannian spaces admitting non-trivial geodesic mappings. A symmetric pair of pseudo-Riemannian spaces is a pair of spaces with coinciding values of covariant derivatives for their Riemann tensors. It is proved that the symmetric pair of pseudo-Riemannian spaces, which are not spaces of constant curvatures, are defined unequivocally by their geodesic lines. The research is carried out locally, using tensors, with no restrictions to the sign of the metric tensor and the signature of a space.

On geodesic mappings of threesymmetric spaces

“…In the theory of geodesic mappings, the main results for these spaces were obtained by M. S. Sinyukov [13]. Later, it became clear that the question of covariant stability of not only the internal objects of pseudo-Riemannian spaces, but also of arbitrary tensors is of interest [3][4][5].…”

“…In the theory of geodesic mappings, the main results for these spaces were obtained by M. S. Sinyukov [13]. Later, it became clear that the question of covariant stability of not only the internal objects of pseudo-Riemannian spaces, but also of arbitrary tensors is of interest [3][4][5].In particular, to study the possibility of reducing the metric tensor to a special form [2]. Following the way of increasing the number of derivatives, M. S. Sinyukov came to the study of geodesic mappings of semisymmetric spaces.…”

Conformal recurrent Kӓhler spaces

“…The notion of tensor recurrence generalizes the notion of symmetry. Other ways of generalization proposed in [8,9,15,21] allow application in the theory of recurrence tensors in Kähler spaces.…”