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Counter-example to the ”second Singer’s theorem”. (English) Zbl 0736.53047

A theorem of I. M. Singer [ Commun. Pure Appl. Math. 13, 685–697 (1960; Zbl 0171.42503)]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 \(3n/2\), according to M. Gromov [Partial differential relations. Berlin etc.: Springer-Verlag (1986; Zbl 0651.53001)]. This theorem raises certain natural questions. In particular one 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 and the paper under review gives a nice counter-example by showing that it cannot hold without an additional topological condition of closeness (for a proof with this additional condition see L. Nicolodi and the reviewer [Ann. Global Anal. Geom. 8, No. 2, 193–209 (1990; 676.53058)].

MSC:

53C30 Differential geometry of homogeneous manifolds
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References:

[1] Kowalski, O.: Generalized Symmetric Spaces, Lecture Notes in Mathematics, Vol. 805, Springer-Verlag 1980. · Zbl 0431.53042
[2] Nicolodi, L., Tricerri, F., On two theorems of I. M. Singer about homogeneous spaces, Ann. Global Anal. Geom. 8 (1990), 193-209. · Zbl 0676.53058 · doi:10.1007/BF00128003
[3] Singer, I. M., Infinitesimally homogeneous spaces,Comm. Pure Appl. Math., 13 (1960), 685-697. · Zbl 0171.42503 · doi:10.1002/cpa.3160130408
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