The theory of F-planar maps of Riemannian spaces and affinely connected spaces developed by J. Mikeš and N. S. Sinyukov [1–6] naturally extends the theory of geodesic and holomorphic projective maps. In the present paper we find basic equations of infinitesimal F-planar maps and study these equations. The F-planar maps are maps between spaces endowed with affinor structures. The geometry of Riemannian spaces and affinely connected spaces endowed by affinor structures was investigated by A. P. Shirokov (see, e.g., [7–14]) who also studied maps between spaces of this type ([13, 14]).

In the present paper we obtain a new proof of the fundamental equations of F-planar mappings of manifolds with affine connections. We discuss alternative ways in the definition of F-planar mappings.

In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function $\psi$ the metric ${\psi }^{-2}g$ is also Einstein, then $\psi$ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel).