The existence of invariant Einstein metrics on a compact homogeneous space
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M. M. Graev
Translated by: E. Khukhro - Trans. Moscow Math. Soc. 2012, 1-28
- DOI: https://doi.org/10.1090/S0077-1554-2013-00202-2
- Published electronically: January 24, 2013
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.References
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Bibliographic Information
- M. M. Graev
- Affiliation: Scientific Research Institute of System Studies of the Russian Academy of Sciences
- Email: mmgraev@gmail.com
- Published electronically: January 24, 2013
- Additional Notes: This research was supported by the Russian Foundation for Basic Research (grant no. 10-01-00041a).
- © Copyright 2013 M. M. Graev
- Journal: Trans. Moscow Math. Soc. 2012, 1-28
- MSC (2010): Primary 53C25; Secondary 53C30
- DOI: https://doi.org/10.1090/S0077-1554-2013-00202-2
- MathSciNet review: 3184965