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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
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