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.


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


[1] Calvaruso, Beiträge Algebra Geom.
[2] Boeckx, Riemannian manifolds of conullity two (1996) · Zbl 0904.53006
[3] DOI: 10.1016/0926-2245(94)90008-6 · Zbl 0796.53046
[4] Boeckx, Arch. Math. (Brno) 29 pp 235– (1993)
[5] Szabó, J. Differential Geom. 17 pp 531– (1982)
[6] Calvaruso, Z. Anal. Anwendungen
[7] Kobayashi, Foundations of differential geometry I, II (1963) · Zbl 0119.37502
[8] Kowalski, J. Math. Pure Appl. 71 pp 471– (1992)
[9] DOI: 10.1007/BF02395060 · Zbl 0428.53017
[10] García-Río, Balkan J. Geom. Appl.
[11] Ferrarotti, Math. J. Toyama Univ. 14 pp 89– (1991)
[12] DOI: 10.1007/BF01214716 · Zbl 0476.53023
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