## Four-dimensional ball-homogeneous and $$C$$-spaces.(English)Zbl 0884.53037

A ball-homogeneous space is a Riemannian manifold on which the volume of a small geodesic ball depends only on the radius. A $$C$$-space is a Riemannian manifold on which the Jacobi operator along each geodesic has constant eigenvalues. We select here two of the main results: (1) A ball-homogeneous four-dimensional Hermitian Einstein space is locally symmetric. (2) A four-dimensional compact Hermitian Einstein $$C$$-space is locally symmetric.

### MSC:

 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces

### Keywords:

4-manifolds; $$C$$-space; ball-homogeneous space; Einstein space
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