Necessary and sufficient conditions for equitorsion geodesic mapping

Abstract

In the present paper we study equitorsion geodesic mappings between two generalized Riemannian spaces f:GRN→GR‾N. In this case these spaces have the same torsions at corresponding points. We prove that a generalized Riemannian space GRN admits an equitorsion geodesic mapping onto a generalized Riemannian space GR‾N if and only if in GRN certain differential equations of Cauchy type in covariant derivatives of the θ=1,…,4 kinds have a solution with respect to the unknown tensor aij, the gradient vector λi≠0, and the differentiable function μθ, θ=1,…,4. In fact we find four systems of PDE all equivalent to the existence of an equitorsion geodesic mapping and discuss the number of linearly independent solutions of this system of PDE. We establish an upper bound for the number of solutions for the geometrical problem under consideration.