On Geodesic Homogeneous Left Lie Loops as Reductive Homogeneous Spaces

Two kinds of canonical connections have been introduced by the author for homogeneous (left) Lie loops, one of which is the canonical connection of the reductive homogeneous spaces induced from the homogeneous (left) Lie loops and the other is the canonical connection of the homogeneous systems associated with left Lie loops. In this paper, it is shown that they are coincident with each other. Moreover, the concept of geodesic reductive homogeneous spaces are introduced. It is shown that the reductive homogeneous space induced from a homogeneous left Lie loop is geodesic if and only if the left Lie loop is geodesic.