Abstract
We introduce a class of Riemann structures, called generalized Einstein structures of index 2e, of which Einstein spaces are a particular case. We show that these structures are stationary for functions introduced on a family of Riemann structures of the compact manifold of H. Weyl. This result solves a problem of M. Berger. As examples of structures which are generalized Einstein structures over all indices we cite homogeneous compact Riemann spaces with a nondecomposable isotropy group and products of such spaces.
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M. Berger, “Quelques formules de variation pour une structure Riemannienne,” Ann. Sci. Ecole Norm. Sup.,3, 285–294 (1970).
H. Weyl, “On the volume of tubes,” Amer. J. Math.,61, 461–472 (1939).
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Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 619–622, October, 1974.
The author thanks A. M. Vasil'ev for a useful discussion of the problems touched upon in this paper.
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Kuz'mina, G.M. Some generalizations of the Riemann spaces of Einstein. Mathematical Notes of the Academy of Sciences of the USSR 16, 961–963 (1974). https://doi.org/10.1007/BF01104264
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DOI: https://doi.org/10.1007/BF01104264