Abstract
We give an explicit example showing that a theorem by I. M. Singer announced in [3] (about the existence of a Riemannian homogeneous space with the prescribed curvature tensor and some of its covariant derivatives) cannot hold without an additional topological condition of closeness.
All references in this short note concern Chapter 3 of the paper by L. Nicolodi and F. Tricerri [2] published in the same volume. We shall use freely the concepts and formulas from there.
Consider the infinitesimal model (V,T,K) given as follows: LetV be a 5-dimensional
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References
Kowalski, O.: Generalized Symmetric Spaces, Lecture Notes in Mathematics, Vol. 805, Springer-Verlag 1980.
Nicolodi, L., Tricerri, F., On two theorems of I. M. Singer about homogeneous spaces, Ann. Global Anal. Geom. 8 (1990), 193–209.
Singer, I. M., Infinitesimally homogeneous spaces,Comm. Pure Appl. Math., 13 (1960), 685–697.
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Kowalski, O. Counter-example to the “Second Singer's theorem”. Ann Glob Anal Geom 8, 211–214 (1990). https://doi.org/10.1007/BF00128004
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DOI: https://doi.org/10.1007/BF00128004