How many are projectable classical linear connections with a prescribed Ricci tensor

Abstract

How many are projectable classical linear connections with
a prescribed Ricci tensor and a prescribed trace of torsion
tensor on the total space of a fibered manifold? The questions are
answered in the analytic case by using the Cauchy-Kowalevski
theorem. In the $C^\infty$ case, we answer
how many are classical linear connections with a prescribed
Ricci tensor on a $2$-dimensional
manifold. In the $C^\infty$ case, we also deduce that any $2$-form
on the total space of a fibered manifold with at least $2$-dimensional fibres
can be realized locally as the Ricci tensor of a projectable classical
linear connection.