On Fε2-planar mappings with function ε of (pseudo-)Riemannian manifolds

Abstract

In this paper we study special mappings between n-dimensional (pseudo-) Riemannian manifolds. In 2003 Topalov introduced PQ?-projectivity of Riemannian metrics, with constant ? ? 0,1 + n. These mappings were studied later by Matveev and Rosemann and they found that for ? = 0 they are projective. These mappings could be generalized for case, when ? will be a function on manifold. We show that PQ?- projective equivalence with ? is a function corresponds to a special case of F-planar mapping, studied by Mikes and Sinyukov (1983) with F = Q. Moreover, the tensor P is derived from the tensor Q and non-zero function ?. We assume that studied mappings will be also F2-planar (Mikes 1994). This is the reason, why we suggest to rename PQ? mapping as F?2. For these mappings we find the fundamental partial differential equations in closed linear Cauchy type form and we obtain new results for initial conditions.