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A remark on locally homogeneous Riemannian spaces. (English) Zbl 0795.53016
Recently, examples have been found of locally homogeneous Riemannian spaces which are not locally isometric to any globally homogeneous Riemannian space. The main result of this paper is the following one: Every locally homogeneous Riemannian space with nonpositive Ricci curvature can be extended to a globally homogeneous Riemannian space. As a consequence, one obtains the following nontrivial generalization of a theorem by Alekseevskij and Kimel’feld: Any locally homogeneous Riemannian manifold with zero Ricci curvature is locally Euclidean.
Reviewer: O.Kowalski (Praha)

53B20 Local Riemannian geometry
53C30 Differential geometry of homogeneous manifolds
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[1] D. V. Alekseevski B. N. Kimel’ Pel’ D, Structure of homogeneous Riemann spaces with zero Ricci curvature, Funktsional’nyi Analiz i Ego Prolozheniya, Vol. 9, n. 2 (1975), 5–11.
[2] L. Berard Bergery, Sur certaines fibrations d’espaces homogene riemanniens, Compositio Math. 30 (1975), 43–61. · Zbl 0304.53036
[3] A. Besse, Einstein Manifolds, Springer-Verlag, Berlin Heidelberg 1987.
[4] N. Bourbaki, Éléments de Mathématique - Groupes et Algébres de Lie, Ch. 2 et 3, Hermann, Paris 1972.
[5] M. Gromov, Partial differential relations, Springer-Verlag 1986. · Zbl 0651.53001
[6] O. Kowalski, Counter-example to the ”Second Singer’s theorem”, Ann. Global Anal. Geometry, 8 (1990). · Zbl 0736.53047
[7] F. Lastaria F. Tricerri, Curvature-orbits and locally homogeneous Riemannian manifolds, Ann. Mat. pura e appl., to appear. · Zbl 0804.53072
[8] J. Milnor, Curvatures of Left Invariant Metrics on Lie Groups, Adv. Math. 21, (1976), 293–329. · Zbl 0341.53030 · doi:10.1016/S0001-8708(76)80002-3
[9] G. D. Mostow, The extensibility of local Lie groups of transformations and groups on surfaces, Ann. of Math. 52 (1950), 606–636. · Zbl 0040.15204 · doi:10.2307/1969437
[10] K. Nomizu, On local and global existence of Killing vector fields, Ann. Math. 76 (1960), 105–120. · Zbl 0093.35103 · doi:10.2307/1970148
[11] A. L. Onishchik E. B. Vinberg, Lie groups and Algebraic groups, Springer-Verlag, Berlin Heidelberg 1990. · Zbl 0722.22004
[12] A. Spiro, Lie pseudogroups and locally homogeneous Riemannian spaces, Boll. U.M.I., 6-B (1992), 843–872. · Zbl 0772.53038
[13] A. Spiro, Sulle varietà Riemanniane localmente omogenee, Tesi di Laurea in Matematica, Università di Camerino Italy, 1991.
[14] F. Tricerri L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Notes, Series 83, Cambridge Univ. Press, Cambridge (1983). · Zbl 0509.53043
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