Two different structures on a manifold (Q1907295)

It is a standard and commonly used approach to consider two different differentiable structures which are related a priori in some sort (projectively, conformally, etc.) and to study the relationship of the geometric objects (connection, curvature, etc.) associated to one structure and those associated to the other structure. The author here proceeds differently. He considers a Finsler structure and a Riemannian structure on a differentiable manifold \(M\) with no relation assumed between the two structures. The Finsler structure induces a (Cartan) connection on the induced bundle \(Q^{-1}(TM)\) and the Riemannian structure induces a (Riemannian) connection on \(M\). Using the notion of lift and projection the author studies the tensor on \(Q^{-1}(TM)\) expressing the difference between the two connections. He investigates the (necessary and sufficient) conditions, to be satisfied by this tensor, for the geometric objects associated to one structure to have the same properties as those associated to the other structure. Among others, the author considers the geometric properties of being a geodesic, a Berwald (Landsberg, locally Minkowskian) space, a Jacobi field, a conservative vector field and of being an infinitesimal transformation. Throughout the paper the author applies the results obtained to the interesting case of a Randers space.