Moscow Mathematical Journal

In the first part of the paper we model an abstract version of Sabinin's theory on transitive families 𝒮 of diffeomorphisms on a differentiable manifold; in particular we define an abstract holonomy group. In the second part we determine the linear connection associated with a smooth family 𝒮 and clarify the relations between it and the properties of 𝒮. Moreover, we prove that all natural holonomy groups are isomorphic if 𝒮 is a geodesic system. Finally we show that the group 𝒜 of smooth automorphisms of 𝒮 is a Lie subgroup of the group of affine transformation of the underlying manifold of 𝒮; if 𝒜 acts transitively we enlighten how 𝒜 influences the algebraic as well as the differential geometric properties of 𝒮.

2010 Mathematics Subject Classification. 20N05, 20B99, 22F50, 20N10