## 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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### References:

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