Geodesic orbit and weakly symmetric spray manifolds

Abstract

In this paper, we introduce the geodesic orbit and weakly symmetric properties in homogeneous spray geometry. When a homogeneous spray manifold is endowed with a reductive decomposition, we use the spray vector field to describe these properties, and prove that a weakly symmetric spray manifold must be geodesic orbit, which generalizes its analog in homogeneous Riemannian and Finsler geometries.

Keywords

• Homogeneous geodesic
• homogeneous spray manifold
• weakly symmetric space
• geodesic orbit space
• spray structure
• spray vector field