Periodica Mathematica Hungarica Vol. 51 (1), 2005, pp. 37–51 INVARIANT SHEN CONNECTIONS AND GEODESIC ORBIT SPACES Zolt´ an Muzsnay (Debrecen) and P´ eter T. Nagy (Debrecen) [Communicated by: J´anos Szenthe] Dedicated to the 65 th birthday of Professor Joseph Grifone Abstract The geodesic graph of Riemannian spaces all geodesics of which are orbits of 1-parameter isometry groups was 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 was 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 investigate a connection theoretical analogue of the concept of the geodesic graph. 1. Introduction Let M = G/H be a homogeneous space equipped with an invariant connec- tion ∇. Let g and h denote the Lie algebra of the Lie group G and H , respectively. The space (M = G/H, ∇) is called affine reductive if there exists an Ad H invariant decomposition g = h + m such that any geodesic γ (t) emanating from the origin o = H ∈ M is the orbit of a 1-parameter subgroup {exp tX, t ∈ R} of G, where X ∈ m, and the parallel translation τ γ 0,t : T γ(0) M → T γ(t) M along the geodesic γ (t) is the same as the left translation by the 1-parameter subgroup {exp tX, t ∈ R}. (cf. [6]). A homogeneous manifold M = G/H with an invariant connection ∇ is called affine geodesic orbit space (g.o. space) if it has the more general property: each geodesic of M is an orbit of a one-parameter subgroup exp tZ (t ∈ R),Z ∈ g. Mathematics subject classification number: 53C05, 53C22, 53C30, 53C60. Key words and phrases: connections, geodesics, homogeneous manifolds, Finsler spaces. This research was partially supported by the Hungarian Foundation for Scientific Research under Grant TO 43516 and by the Hungarian Higher Education, Research and Development Fund (FKFP) Grant 0184/2001. 0031-5303/2005/$20.00 Akad´ emiai Kiad´o, Budapest c  Akad´ emiai Kiad´o, Budapest Springer, Dordrecht