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Transactions of the Moscow Mathematical Society

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The existence of invariant Einstein metrics on a compact homogeneous space

Author: M. M. Graev
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 1-28
MSC (2010): Primary 53C25; Secondary 53C30
Published electronically: January 24, 2013
MathSciNet review: 3184965
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Abstract: It is proved that if the triangularizable compact set $ C=X_{G,H}^{\Sigma }$ introduced by Böhm (Böhm's polyhedron) is non-contractible, then there exists a positive-definite invariant Einstein metric $ m$ of positive scalar curvature on a connected homogeneous space $ G/H$ of a compact Lie group $ G$. There is a natural continuous map of $ C$ onto the flag complex $ K_{B}$ of a finite graph $ B$. For $ C=K_B$ this gives one of the criteria proved by Böhm. Another consequence -- that $ m$ exists for disconnected $ B$ -- is a version of the Böhm-Wang-Ziller graph theorem (but now the graph may be disconnected for $ \mathfrak{z(g)}\ne 0$). Furthermore, the preparatory Böhm theorems on retractions are revised, and in this connection new constructions of certain topological spaces are proposed.

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Additional Information

M. M. Graev
Affiliation: Scientific Research Institute of System Studies of the Russian Academy of Sciences

Keywords: Homogeneous manifold, Einstein metric
Published electronically: January 24, 2013
Additional Notes: This research was supported by the Russian Foundation for Basic Research (grant no. 10-01-00041a).
Article copyright: © Copyright 2013 M. M. Graev

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