Invariant Shen connections and geodesic orbit spaces

The geodesic graph of Riemannian spaces all geodesics of which are orbits of 1-parameter isometry groups is constructed by J. Szenthe in 1976 and it became a basic tool for studying such spaces, called g.o. spaces. This infinitesimal structure corresponds to the reductive complement m in the case of naturally reductive spaces. The systematic study of Riemannian g.o. spaces is started by O. Kowalski and L. Vanhecke in 1991, when they introduced the most important definitions, classified the low-dimensional examples and described the basic constructions of this theory. The aim of this paper is to introduce a connection theoretical version of the concept of the geodesic graph. This type of connection has been considered in an early paper of L. Berwald and it is strongly related to the Finsler connection theory of S. S. Chern and of H. Rund. We give a global treatment of generalized linear connections which are used systematically by Z. Shen for the investigation of Finsler manifolds.