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Semi-symmetric ball-homogeneous spaces and a volume conjecture. (English) Zbl 0903.53031

A Riemannian space \((M,g)\) is ball-homogeneous if the volume of any sufficiently small geodesic ball depends only on its radius. It is still an open problem whether any ball-homogeneous space is locally homogeneous. In the present article, the authors give a positive answer when \((M,g)\) is semi-symmetric (i.e., the Riemann curvature tensor of \((M,g)\) is the same as that of a symmetric space at each point). The space is then even locally symmetric. As a consequence, a semi-symmetric space such that the volume of all sufficiently small geodesic balls is the same as in Euclidean space is locally flat.

MSC:

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