Homogeneous Finsler Spaces

Abstract

In this chapter we study homogeneous Finsler spaces. In Sect. 4.1, we define the notions of Minkowski Lie pairs and Minkowski Lie algebras to give an algebraic description of invariant Finsler metrics on homogeneous manifolds and bi-invariant Finsler metrics on Lie groups. Then in Sect. 4.2, we present a sufficient and necessary condition for a coset space to have invariant non-Riemannian Finsler metrics. In Sect. 4.3, we study homogeneous Finsler spaces of negative curvature and prove that every homogeneous Finsler space with nonpositive flag curvature and negative Ricci scalar must be simply connected. In Sect. 4.4, we apply our result to study the degree of symmetry of closed manifolds. In particular, we prove that if a closed manifold is not diffeomorphic to a rank-one Riemannian symmetric space, then its degree of symmetry can be realized by a non-Riemannian Finsler metric. Finally, in Sect. 4.5, we study fourth-root homogeneous Finsler metrics. As an explicit example, we give a classification of all invariant fourth-root Finsler metrics on Grassmannian manifolds.