Submersion and Homogeneous Spray Geometry

We introduce the submersion between two spray structures and propose technique in geometry. Using this technique, as well global invariant frames on a Lie group, we setup general theoretical framework for homogeneous define vector field $$\eta $$ connection operator N manifold $$(G/H,{\mathbf {G}})$$ with linear decomposition $${\mathfrak {g}}={\mathfrak {h}}+{\mathfrak {m}}$$ . These notions generalize their counter parts Finsler prove correspondence $${\mathbf {G}}$$ when given is reductive, that geodesics integral curves of $$-\eta find ordinary differential equations describing parallel translations , calculate S-curvature Riemann curvature generalizing L. Huang’s formulae manifolds.