A new approach to generalized Berwald manifolds I

Abstract

A large class of special Finsler manifolds can be endowed with Finsler connections whose “$h$-part” does not depend on the directions. We call these Finsler connections $h$-basic and present a systematic treatment of them, using (in a simplified form) the Frölicher-Nijenhuis calculus. We provide an axiomatic description of a distinguished class of $h$-basic Finsler connections, the class of Ichijyō connections. With the help of an Ichijyō connection we present new characterizations of generalized Berwald manifolds, as well as – in particular – of Berwald manifolds and locally Minkowski manifolds.