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For modelling of various physical processes, geodesic lines and almostgeodesic curves serve as a useful tool. Trasformations or mappings betweenspaces (endowed with a metric or connection) which preserve such curves play animportant role in physics, particularly in mechanics, and in geometry as well.Our aim is to continue investigations concerning existence of almost geodesicmappings of manifolds with linear (affine) connection, particularly of theso-called ${\tilde \pi}_1$ mappings, i.e. canonical almost geodesic mappings oftype $\pi_1$ according to Sinyukov. First we give necessary and sufficientconditions for existence of ${\tilde \pi}_1$ mappings of a manifold endowedwith a linear connection onto pseudo-Riemannian manifolds. The conditions takethe form of a closed system of PDE's of first order of Cauchy type. Further wededuce necessary and sufficient conditions for existence of ${\tilde \pi}_1$mappings onto generalized Ricci-symmetric spaces. Our results aregeneralizations of some previous theorems obtained by N.S. Sinyukov.

Fundamental PDE's of the Canonical Almost Geodesic Mappings of Type ${\tilde\pi}_1$