Ball-homogeneous spaces. (English) Zbl 0912.53034

Calvaruso, G.; Vanhecke, Lieven

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].

Reviewer: Oldřich Kowalski (Praha)

MSC:

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

Keywords:

ball-homogeneous space; conformally flat; locally symmetric; Ricci eigenvalues; Kähler-Einstein space