Abstract

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

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