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INVARIANT EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH TWO ISOTROPY SUMMANDS

Part of: Global differential geometry Lie groups

Published online by Cambridge University Press:  19 July 2011

ANDREAS ARVANITOYEORGOS  and
IOANNIS CHRYSIKOS
ANDREAS ARVANITOYEORGOS
Affiliation:
Department of Mathematics, University of Patras, GR-26500 Rion, Greece (email: arvanito@math.upatras.gr)
IOANNIS CHRYSIKOS*
Affiliation:
Department of Mathematics, University of Patras, GR-26500 Rion, Greece (email: xrysikos@master.math.upatras.gr)
*
✉For correspondence; e-mail: xrysikos@master.math.upatras.gr
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Abstract

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Let M=G/K be a generalized flag manifold, that is, an adjoint orbit of a compact, connected and semisimple Lie group G. We use a variational approach to find non-Kähler homogeneous Einstein metrics for flag manifolds with two isotropy summands. We also determine the nature of these Einstein metrics as critical points of the scalar curvature functional under fixed volume.

Keywords

Einstein manifoldhomogeneous spacegeneralized flag manifoldisotropy representationhighest weightWeyl’s formulabordered Hessian

MSC classification

Secondary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.) 22E46: Semisimple Lie groups and their representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 90 , Issue 2 , April 2011 , pp. 237 - 251
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The authors were partially supported by C. Carathéodory grant number C.161 2007-10, University of Patras.

References

[Ale]Alekseevsky, D. V., ‘Homogeneous Einstein metrics’, in: Differential Geometry and its Applications (Proceedings of the Conference) (J. E. Purkyně University, Brno, Czechoslovakia, 1987), pp. 1–21.Google Scholar
[AlA]Alekseevsky, D. V. and Arvanitoyeorgos, A., ‘Riemannian flag manifolds with homogeneous geodesics’, Trans. Amer. Math. Soc. 359 (2007), 3769–3789.Google Scholar
[Arv]Arvanitoyeorgos, A., ‘New invariant Einstein metrics on generalized flag manifolds’, Trans. Amer. Math. Soc. 337 (1993), 981–995.Google Scholar
[ArC]Arvanitoyeorgos, A. and Chrysikos, I., ‘Motion of charged particles and homogeneous geodesics in Kähler C-spaces with two isotropy summands’, Tokyo J. Math. 32 (2009), 487–500.Google Scholar
[Bes]Besse, A. L., Einstein Manifolds (Springer, Berlin, 1986).Google Scholar
[Bom]Böhm, C., ‘Homogeneous Einstein metrics and simplicial complexes’, J. Differential Geom. 67 (2004), 79–165.Google Scholar
[BWZ]Böhm, C., Wang, M. and Ziller, W., ‘A variational approach for compact homogeneous Einstein manifolds’, Geom. Funct. Anal. 14 (2004), 681–733.CrossRefGoogle Scholar
[BFR]Borderman, M., Forger, M. and Römer, H., ‘Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models’, Comm. Math. Phys. 102 (1986), 604–647.Google Scholar
[Bor]Borel, A., ‘Kählerian coset spaces of semi-simple Lie groups’, Proc. Natl. Acad. Sci. USA 40 (1954), 1147–1151.Google Scholar
[BoH]Borel, A. and Hirzebruch, F., ‘Characteristic classes and homogeneous spaces I’, Amer. J. Math. 80 (1958), 458–538.CrossRefGoogle Scholar
[DKe]Dickinson, W. and Kerr, M. M., ‘The geometry of compact homogeneous spaces with two isotropy summands’, Ann. Global Anal. Geom. 34 (2008), 329–350.CrossRefGoogle Scholar
[GOV]Gorbatzevich, V. V., Onishchik, A. L. and Vinberg, E. B., Lie groups and Lie algebras. III. Structure of Lie Groups and Lie Algebras, Encyclopedia of Mathematical Sciences, 41 (Springer, Berlin, 1994).CrossRefGoogle Scholar
[Hel]Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces (Academic Press, New York, 1978).Google Scholar
[Ikw]Ikawa, O., ‘Motion of charged particles in Kähler C-spaces’, Yokohama Math. J. 50 (2003), 31–39.Google Scholar
[Ith]Itoh, M., ‘On curvature properties of Kähler C-spaces’, J. Math. Soc. Japan 30 (1978), 39–71.CrossRefGoogle Scholar
[Kim]Kimura, M., ‘Homogeneous Einstein metrics on certain Kähler C-spaces’, Adv. Stud. Pure Math. 18-I (1990), 303–320.Google Scholar
[KoN]Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Vol. II (Wiley-Interscience, New York, 1969).Google Scholar
[LWa] C. LeBrun and M. Wang (eds), Surveys in Dierential Geometry Volume VI: Essays on Einstein Manifolds (International Press, Somerville, MA, 1999).Google Scholar
[MaT]Marsden, J. E. and Tromba, A. J., Vector Calculus, 5th edn (W. H. Freeman, New York, 2003).Google Scholar
[NRS]Nikonorov, Yu. G., Rodionov, E. D. and Slavskii, V. V., ‘Geometry of homogeneous Riemannian manifolds’, J. Math. Sci. 146 (2007), 6313–6390.Google Scholar
[Ohm]Ohmura, I., ‘On Einstein metrics on certain homogeneous spaces’, master’s thesis (Japanese), Graduate School of Science, Osaka University, 1987 (unpublished).Google Scholar
[Sak]Sakane, Y., ‘Homogeneous Einstein metrics on flag manifolds’, Lobachevskii J. Math. 4 (1999), 71–87.Google Scholar
[Wan]Wang, H. C., ‘Closed manifolds with homogeneous complex structures’, Amer. J. Math. 76 (1954), 1–32.CrossRefGoogle Scholar
[WZ1]Wang, M. and Ziller, W., ‘On normal homogeneous Einstein manifolds’, Ann. Sci. Éc. Norm. Supér. 18 (1985), 563–633.CrossRefGoogle Scholar
[WZ2]Wang, M. and Ziller, W., ‘Existence and non-existence of homogeneous Einstein metrics’, Invent. Math. 84 (1986), 177–194.CrossRefGoogle Scholar
[Wol]Wolf, J. A., ‘The action of a real semi-simple Lie group on a complex flag manifold, I: orbit structure and holomorphic arc components’, Bull. Amer. Math. Soc. 75 (1969), 1121–1237.CrossRefGoogle Scholar
[Zil]Ziller, W., ‘Homogeneous Einstein metrics on spheres and projective spaces’, Math. Ann. 259 (1982), 351–358.Google Scholar