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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
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