Abstract
The classification of compact homogeneous spaces of the form \(M=G/K\), where G is a non-simple Lie group, such that the standard metric is Einstein is still open. The only known examples are 4 infinite families and 3 isolated spaces found by Nikonorov and Rodionov in the 90 s. In this paper, we prove that most of these standard Einstein metrics are unstable as critical points of the scalar curvature functional on the manifold of all unit volume G-invariant metrics on M, providing a lower bound for the coindex in the case of Ledger–Obata spaces. On the other hand, examples of stable (in particular, local maxima) invariant Einstein metrics on certain homogeneous spaces of non-simple Lie groups are also given.
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References
Chen, Z., Nikonorov, Yu.G., Nikonorova, Y.V.: Invariant Einstein metrics on Ledger–Obata spaces. Diff. Geom. App. 50, 7–87 (2017)
D’Atri, J., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups, Mem. Amer. Math. Soc. 215 (1979)
Jensen, G.: Einstein metrics on principal fibre bundles. J. Diff. Geom. 8, 599–614 (1973)
Lauret, E.A., Lauret, J.: The stability of standard homogeneous Einstein manifolds. Math. Z. 303, 16 (2023)
Lauret, J.: On the stability of homogeneous Einstein manifolds. Asian J. Math. 26, 555–584 (2022)
Lauret, J., Will, C.E.: On the stability of homogeneous Einstein manifolds II. J. London Math. Soc. 106, 3638–3669 (2022)
Nikonorov, Yu.G.: Algebraic structure of standard homogeneous Einstein manifolds. Siberian Adv. Math. 10, 59–82 (2000)
Nikonorov, Yu.G., Rodionov, E.D.: Standard homogeneous Einstein manifolds and Diophantine equations. Archivum. Math. (Brno) 32, 123–136 (1996)
Nikonorov, Yu.G., Rodionov, E.D., Slavskii, V.V.: Geometry of homogeneous Riemannian manifolds. J. Math. Sci. (N.Y.) 146, 6313–6390 (2007)
Wang, M.Y., Ziller, W.: On normal homogeneous Einstein manifolds. Ann. Sci. École Norm. Sup. 18, 563–633 (1985)
Acknowledgements
The authors thank Yuri Nikonorov for very helpful conversations on the subject of this paper and the referee for very helpful suggestions.
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This research was partially supported by grants from FONCyT and Univ. Nac. de Córdoba.
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Gutiérrez, V., Lauret, J. Stability of standard Einstein metrics on homogeneous spaces of non-simple Lie groups. Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00431-7
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DOI: https://doi.org/10.1007/s13348-023-00431-7