On homogeneous Riemannian manifolds

Infinitesimally homogeneous spaces

Riemannian-manifolds with homogeneous geodesics

Publisher Summary This chapter discusses the theory of Hausdorff convergence. Classically, the Hausdorff distance between two closed subsets in a fixed metric space has been defined. Gromov defined the Hausdorff distance between two metric spaces by taking the infimum of the Hausdorff distances over all ambient spaces into which the two metric spaces are embedded by isometries. There he proved two fundamental results—the precompactness theorem and the convergence theorem—and applied them to the study of the global structures of Riemannian manifolds. The chapter also discusses generalizations of Hausdorff convergence to the case of open manifolds and/or manifolds on which groups act.

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Hausdorff Convergence of Riemannian Manifolds and Its Applications

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.

On homogeneous Riemannian manifolds

Citations

Infinitesimally homogeneous spaces

Riemannian-manifolds with homogeneous geodesics

Hausdorff Convergence of Riemannian Manifolds and Its Applications

Homogeneous Finsler Spaces

Homogeneous structures on three-dimensional Lorentzian manifolds

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