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Semi-symmetric ball-homogeneous spaces and a volume conjecture

Published online by Cambridge University Press:  17 April 2009

G. Calvaruso  and
L. Vanhecke
G. Calvaruso
Affiliation:
Department of MathematicsKatholieke Universiteit LeuvenCelestijnenlaas 200 B3001 LeuvenBelgium
L. Vanhecke
Affiliation:
Department of MathematicsKatholieke Universiteit LeuvenCelestijnenlaas 200 B3001 LeuvenBelgium
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Abstract

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We prove that semi-symmetric ball-homogeneous spaces are locally symmetric and we use this result to prove that a semi-symmetric Riemannian manifold such that the volume of each sufficiently small geodesic ball is the same as in a Euclidean space, is locally flat.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 1 , February 1998 , pp. 109 - 115
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Boeckx, E., ‘Einstein-like semi-symmetric spaces’, Arch. Math. (Brno) 29 (1993), 235–240.Google Scholar
[2]Boeckx, E., Kowalski, O. and Vanhecke, L., ‘Non-homogeneous relatives of symmetric spaces’, Differential Geom. Appl. 4 (1994), 45–69.CrossRefGoogle Scholar
[3]Boeckx, E., Kowalski, O. and Vanhecke, L., Riemannian manifolds of conullity two (World Scientific Publ. Co., Singapore, 1996).CrossRefGoogle Scholar
[4]Calvaruso, G., Tondeur, Ph. and Vanhecke, L., ‘Four-dimensional ball-homogeneous and C-spaces’, Beiträge Algebra Geom. (to appear).Google Scholar
[5]Calvaruso, G. and Vanhecke, L., ‘Special ball-homogeneous spaces’, Z. Anal. Anwendungen (to appear).Google Scholar
[6]Ferrarotti, M. and Vanhecke, L., ‘Locally symmetric spaces and the volume conjecture for geodesic balls’, Math. J. Toyama Univ. 14 (1991), 89–93.Google Scholar
[7]García-Río, E. and Vanhecke, L., ‘Five-dimensional φ-symmetric spaces’, Balkan J. Geom. Appl. (to appear).Google Scholar
[8]Gray, A. and Vanhecke, L., ‘Riemannian geometry as determined by the volumes of small geodesic balls’, Acta Math. 142 (1979), 157–198.Google Scholar
[9]Kobayashi, S. and Nomizu, K., Foundations of differential geometry I, II (Interscience Publ., New York, 1963, 1969).Google Scholar
[10]Kowalski, O., Tricerri, F. and Vanhecke, L., ‘Curvature homogeneous Riemannian manifolds’, J. Math. Pure Appl. 71 (1992), 471–501.Google Scholar
[11]Kowalski, O. and Vanhecke, L., ‘Ball-homogeneous and disk-homogeneous Riemannian manifolds’, Math. Z. 180 (1982), 429–444.CrossRefGoogle Scholar
[12]Szabó, Z.I., ‘Structure theorems on Riemannian manifolds satisfying R(X, Y). R = 0, I, the local version’, J. Differential Geom. 17 (1982), 531–582.CrossRefGoogle Scholar
[13]Vanhecke, L., ‘Scalar curvature invariants and local homogeneity’, in Proc. Workshop on Differential Geometry and Topology (dedicated to the memory of Franco Tricerri) Palermo 1996 (to appear).Google Scholar