On quasi-geodesic mappings of special pseudo-Riemannian spaces

Abstract

The present paper continues the study of quasi-geodesic mappings f:(Vn, gij, Fih) → (V'n,g'ij, Fih) of pseudo-Riemannian spaces Vn, V'n with a generalized-recurrent structure Fih of parabolic type. By a generalized recurrent structure of parabolic type on Vn we mean an almost Hermitian affinor structure of parabolic type for which the covariant derivative of the structural affinor Fih satisfies the condition F(i,j)h=q(i Fj)h.
 In the previous paper by the authors [Proc. Intern. Geom. Center, 13:3 (2020) 18-32] it was proved that the class of pseudo-Riemannian spaces with generalized-recurrent structure of parabolic type is closed with respect to the considered mappings and the generalized recurrence vectors in (Vn, gij,Fih) and (V'_n, g'ij, Fih) may be distinct. In this article, it is assumed that the mapping f preserves the generalized recurrence vector qi.
 We construct geometric objects that are invariant under the quasi-geodesic mapping of generalized-recurrent spaces of parabolic type and recurrent-parabolic spaces. A number of conditions are given on these objects, which lead to the fact that a generalized-recurrent space of parabolic type admits a parabolic K-structure, and a recurrent-parabolic space admits a Kählerian structure of parabolic type.
 We study special types of these mappings that preserve some tensors of an intrinsic nature.