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On two theorems of I. M. Singer about homogeneous spaces. (English) Zbl 0676.53058
A theorem of I. M. Singer [Commun. Pure Appl. Math. 13, 685-697 (1960; Zbl 0171.425)] states that a homogeneous Riemannian manifold is determined, up to local isometries, by the values at a fixed point of the Riemann curvature tensor and of its covariant derivatives up to some order $$k_ M$$ depending only on the dimension n of the manifold [not exceeding (3/2)n, according to M. Gromov, Partial differential relations (1986; Zbl 0651.53001), p. 165]. This theorem raises certain natural questions. In particular we can ask how it is possible to recover a homogeneous Riemannian manifold from the Riemann curvature tensor and its covariant derivatives up to the order $$k_ M$$. The answer to this question is given at the end of the quoted paper of I. M. Singer without proof. Actually, the statement of Singer is incomplete, as an example due to O. Kowalski shows [see O. Kowalski, Ann. Global Anal. Geom. (to appear)]. The aim of our paper is to complete the statement of Singer and to supply a simple direct proof.
Reviewer: F.Tricerri

##### MSC:
 53C30 Differential geometry of homogeneous manifolds
Full Text:
##### References:
 [1] W. Ambrose and I. M. Singer, On homogeneous Riemannian manifolds, Duke Math. J.,25 (1958), 647-669. · Zbl 0134.17802 [2] M. Berger, P. Gauduchon, E. Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Math.194, Springer, Berlin 1971. · Zbl 0223.53034 [3] A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 10, Springer, Berlin 1987. · Zbl 0613.53001 [4] M. Gromov, Structures métriques pour les variétés riemanniennes, Cedic, Paris 1981. [5] D. Husemoller, Fibre bundles, second edition, Graduate Texts in Mathematics20, Springer, New York-Heidelberg-Berlin 1975. [6] S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience Publ., New York, vol. I (1963), vol. II (1969). · Zbl 0119.37502 [7] O. Kowalski, Counter-example to the ?Second Singer’s Theorem?, Ann. Global Anal. Geom.,8 (1990), 211-214. · Zbl 0736.53047 [8] K. Nomizu, Invariant affine connections on homogeneous spaces. Amer. J. Math.,76 (1954), 33-65. · Zbl 0059.15805 [9] I. M. Singer, Infinitesimally homogeneous spaces, Comm. Pure Appl. Math.,13 (1960), 685-697. · Zbl 0171.42503 [10] F. Tricerri and L. Vanhecke, Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Notes Series 83, Cambridge Univ. Press, Cambridge 1983. · Zbl 0509.53043 [11] F. Tricerri and L. Vanhecke, Curvature homogeneous Riemannian manifolds, Ann. Sci. Ecole Norm. Sup.,22 (1989), 535-554. · Zbl 0698.53033
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