Ball-homogeneous spaces. (English) Zbl 0912.53034

Cordero, L. A. (ed.) et al., Proceedings of the workshop on recent topics in differential geometry, Santiago de Compostela, Spain, July 16–19, 1997. Santiago de Compostela: Universidade de Santiago de Compostela. Publ. Dep. Geom. Topología, Univ. Santiago Compostela. 89, 35-51 (1998).
A Riemannian manifold \((M,g)\) is said to be ball-homogeneous if the volume of every sufficiently small geodesic ball does not depend on the center of the ball but is only a function of the radius. Locally homogeneous and connected Riemannian manifolds are ball-homogeneous; the convergence is an interesting open problem which is not completely solved even in dimension three.
Here are some selected results. (1) A semi-symmetric ball-homogeneous space is locally homogeneous. (2) A three-dimensional ball-homogeneous space with at most two distinct Ricci eigenvalues is locally homogeneous. (3) A curvature homogeneous conformally flat Riemannian manifold is locally symmetric. (4) An \(n\)-dimensional conformally flat ball-homogeneous space with at most three distinct Ricci eigenvalues is locally symmetric. (5) A four-dimensional ball-homogeneous Kähler-Einstein space is locally symmetric.
For the entire collection see [Zbl 0893.00034].


53C30 Differential geometry of homogeneous manifolds
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53C55 Global differential geometry of Hermitian and Kählerian manifolds