Abstract
A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands by the use of \({\mathfrak{t}}\)-roots. We present new G-invariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akhiezer D.N.: Lie Group Actions in Complex Analysis, Aspects of Mathematics, vol. E27. Vieweg, Braunschweig (1995)
Alekseevsky, D.V.: Homogeneous Einstein metrics. In: Differential Geometry and its Applications (Proccedings of the Conference), pp. 1–21. Univ. of. J. E. Purkyne, Chechoslovakia (1987)
Alekseevsky D.V., Arvanitoyeorgos A.: Riemannian flag manifolds with homogeneous geodesics. Trans. Am. Math. Soc. 359(8), 3769–3789 (2007)
Alekseevsky D.V., Perelomov A.M.: Invariant Kähler-Einstein metrics on compact homogeneous spaces. Funct. Anal. Appl. 20(3), 171–182 (1986)
Alekseevsky, D.V., Spiro, A.F.: Flag Manifolds and Homogeneous CR Structures. Lecture notes given by the first author at the First Colloquium on Lie Theory and Applications in Spain (2000)
Arvanitoyeorgos A.: New invariant Einstein metrics on generalized flag manifolds. Trans. Am. Math. Soc. 337(2), 981–995 (1993)
Arvanitoyeorgos A.: Geometry of flag manifolds. Int. J. Geom. Methods Mod. Phys. 3(5-6), 1–18 (2006)
Arvanitoyeorgos, A., Chrysikos, I.: Invariant Einstein metrics on generalized flag manifolds with two isotropy summands, Preprint (2008), arXiv:0902.1826v1
Arvanitoyeorgos, A., Chrysikos, I.: Motion of charged particles and homogeneous geodesics in Kähler C-spaces with two isotropy summands, Tokyo J. Math. (to appear)
Besse A.L.: Einstein Manifolds. Springer-Verlag, Berlin (1986)
Bordeman M., Forger M., Römer H.: Homogeneous Kähler manifolds: paving the way towards new supersymmetric sigma models. Comm. Math. Phys. 102, 604–647 (1986)
Borel A.: Kahlerian coset spaces of semisimple Lie groups. Proc. Natl Acad. Sci. USA 40, 1147–1151 (1954)
Borel A., Hirzebruch F.: Characteristics classes and homogeneous spaces I. Am. J. Math. 80, 458–538 (1958)
Böhm C., Kerr M.: Low-dimensional homogeneous Einstein manifolds. Trans. Amer. Math. Soc. 358(4), 1455–1468 (2005)
Böhm C., Wang M., Ziller W.: A variational approach for homogeneous Einstein metrics. Geom. Funct. Anal. 14(4), 681–733 (2004)
Bourbaki N.: Éléments De Mathématique: Groupes et algèbres De Lie, Chapitres 4, 5 et 6. Masson Publishing, Paris (1981)
Boyer C.P., Galicki K.: On Sasakian-Einstein geometry. Int. J. Math. 11, 873–909 (2000)
Burstall F.E.: Riemannian twistor spaces and holonomy groups. In: Bailey, T.N., Baston, R.J. (eds) Twistors in Mathematics and Physics, pp. 53–70. Cambridge University Press, Cambridge (1990)
Burstall, F.E., Rawnsley, J.H.: Twistor Theory for Riemannian Symmetric Spaces. Lectures Notes in Mathematics, Springer-Verlag (1990)
Dickinson W., Kerr M.: The geometry of compact homogeneous spaces with two isotropy summands. Ann. Glob. Anal. Geom. 34, 329–350 (2008)
Dos Santos E.C.F., Negreiros C.J.C.: Einstein metrics on flag manifolds. Revista Della, Unión Mathemática Argetina 47(2), 77–84 (2006)
Gorbatzevich, V.V., Onishchik, A.L., Vinberg, E.B.: Structure of Lie Groups and Lie Algebras, Encycl. Math. Sci. v41, Lie Groups and Lie Algebras-3, Springer–Verlag
Graev M.M.: On the number of invariant Einstein metrics on a compact homogeneous space, Newton polytopes and contraction of Lie algebras. Int. J. Geom. Methods Mod. Phys. 3(5–6), 1047–1075 (2006)
Heber J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)
Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York (1978)
Itoh M.: On curvature properties of Kähler C-spaces. J. Math. Soc. Jpn 30(1), 39–71 (1978)
Kimura M.: Homogeneous Einstein metrics on certain Kähler C-spaces. Adv. Stud. Pure Math. 18(I), 303–320 (1990)
LeBrun, C., Wang, M. (eds): Surveys in Differential Geometry, Volume VI: Essays on Einstein Manifolds. International Press, Boston (1999)
Lomshakov, A., Nikonorov, Yu.G., Firsov, E.: Invariant Einstein Metrics on Three-Locally-Symmetric spaces. (Russian) Mat. Tr. 6(2), 80–101 (2003), (engl. transl. in Siberian Adv. Math. 14(3), 43–62 (2004))
Nikonorov Yu.G.: Compact homogeneous Einstein 7-manifolds. Geom. Dedic. 109, 7–30 (2004)
Nikonorov Yu.G., Rodionov E.D., Slavskii V.V.: Geometry of homogeneous Riemannian manifolds. J. Math. Sci. 146(6), 6313–6390 (2007)
Nishiyama M.: Classification of invariant complex structures on irreducible compact simply connected coset spaces. Osaka J. Math. 21, 39–58 (1984)
Park J.-S., Sakane Y.: Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math. 20(1), 51–61 (1997)
Sakane Y.: Homogeneous Einstein metrics on flag manifolds. Lobachevskii J. Math. 4, 71–87 (1999)
Samelson H.: Notes on Lie Algebras. Springer-Verlag, New York (1990)
Siebenthal J.: Sur certains modules dans une algébre de Lie semisimple. Comment. Math. Helv. 44(1), 1–44 (1964)
Tian G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–37 (1997)
Wang H.C.: Closed manifolds with homogeneous complex structures. Am. J. Math. 76(1), 1–32 (1954)
Wang M., Ziller W.: On normal homogeneous Einstein manifolds. Ann. Sci. Éc. Norm. Sup. 18(4), 563–633 (1985)
Wang M., Ziller W.: Existence and non-excistence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Arvanitoyeorgos, A., Chrysikos, I. Invariant Einstein metrics on flag manifolds with four isotropy summands. Ann Glob Anal Geom 37, 185–219 (2010). https://doi.org/10.1007/s10455-009-9183-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-009-9183-7
Keywords
- Homogeneous manifold
- Einstein metric
- Generalized flag manifold
- Isotropy representation
- \({\mathfrak{t}}\)-roots