Ehresmann connections, metrics and good metric derivatives

Abstract

In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an outline of our main tools, i.e., the pull-back bundle formalism, we give an overview of Ehresmann connections and covariant derivatives in the pull-back bundle of a tangent bundle over itself. Then we define and characterize some special classes of generalized metrics. By a generalized metric we shall mean a pseudo-Riemannian metric tensor in our pull-back bundle. The main new results are contained in Section 5. We shall say, informally, that a metric covariant derivative is 'good' if it is related in a natural way to an Ehresmann connection determined by the metric alone. We shall find a family of good metric derivatives for the so-called weakly normal Moór–Vanstone metrics and a distinguished good metric derivative for a certain class of Miron metrics.