Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups

Arvanitoyeorgos, Andreas ; Dzhepko, V. V. ; Nikonorov, Yu. G.

Canadian journal of mathematics, Tome 61 (2009) no. 6, pp. 1201-1213

Résumé

A Riemannian manifold $\left( M,\,\rho\right)$ is called Einstein if the metric $\rho $ satisfies the condition $\text{Ric}\left( \rho\right)\,=\,c\,\cdot \,\rho $ for some constant $c$ . This paper is devoted to the investigation of $G$ -invariant Einstein metrics, with additional symmetries, on some homogeneous spaces $G/H$ of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds $SO\left( n \right)/SO\left( l \right)$ . Furthermore, we show that for any positive integer $p$ there exists a Stiefel manifold $SO\left( n \right)/SO\left( l \right)$ that admits at least $p$ $SO\left( n \right)$ -invariant Einstein metrics.

Détail

DOI : 10.4153/CJM-2009-056-2

Mots-clés : 53C25, 53C30, Riemannian manifolds, homogeneous spaces, Einstein metrics, Stiefel manifolds

Bibliographie

[1] [1] Alekseevsky, D. V., Dotti, I., and Ferraris, C., Homogeneous Ricci positive 5-manifolds, Pacific J. Math. 175(1996), no. 1, 1–12. Google Scholar

[2] [2] Arvanitoyeorgos, A., New invariant Einstein metrics on generalized flag manifolds. Trans. Amer. Math. Soc. 337(1993), no. 2, 981–995. Google Scholar

[3] [3] Back, A. and Hsiang, W. Y., Equivariant geometry and Kervaire spheres. Trans. Amer. Math. Soc. 304(1987) no. 1, 207–227. Google Scholar

[4] [4] Besse, A., Einstein Manifolds. Ergebnisse der mathematik und ihrer Grenzgebiete 10, Springer-Verlag, Berlin, 1987. Google Scholar

[5] [5] Böhm, C., Homogeneous Einstein metrics and simplicial complexes. J. Differential Geom. 67(2004), no. 1, 79–165. Google Scholar

[6] [6] Böhm, C. and Kerr, M., Low-dimensional homogenous Einstein manifolds. Trans. Amer. Math. Soc. 358(2006), no. 4, 1455–1468. Google Scholar

[7] [7] Böhm, C., Wang, M., and Ziller, W., A variational approach for compact homogeneous Einstein manifolds. Geom. Func. Anal. 14(2004), no. 4, 681–733. Google Scholar

[8] [8] D’Atri, J. E. and Nickerson, N., Geodesic symmetries in space with special curvature tensors, J. Diff. Geom. 9 (1974) 251–262. Google Scholar

[9] [9] D’Atri, J. E. and Ziller, W., Naturally reductive metrics and Einstein metrics on compact Lie groups. Memoirs Amer. Math. Soc. 18(1979) no. 215. Google Scholar

[10] [10] Jensen, G., The scalar curvature of left-invariant Riemannian metrics. Indiana J. Math. 20(1971) 1125–1144. Google Scholar

[11] [11] Jensen, G., Einstein metrics on principal fiber bundles J. Differential Geom. 8(1973), 599–614. Google Scholar

[12] [12] Kerr, M., New examples of homogeneous Einstein metrics. Michigan J. Math. 45(1998)no. 1, 115–134. Google Scholar

[13] [13] Kobayashi, S., Topology of positive pinched Kähler manifolds. Tôhoku Math. J. 15(1963), 121–139. Google Scholar

[14] [14] Kimura, M., Homogeneous Einstein metrics on certain Kähler C-spaces. In: Recent Topics in Differential and Analytic Geometry. Adv. Stud. Pure Math. 18-I. Academic Press, Boston, MA, 1990, pp. 303–320. Google Scholar

[15] [15] Lomshakov, A. V., Nikonorov, Yu. G., and Firsov, E. V., Invariant Einstein metrics on three-locally-symmetric spaces. (Russian) Mat. Tr. 6(2003), no. 2, 80–101; Engl. transl. in Siberian Adv. Math. 14(2004), no. 3, 43–62. Google Scholar

[16] [16] Sagle, A., Some homogeneous Einstein manifolds. Nagoya Math. J. 39(1970), 81–106. Google Scholar

[17] [17] Nikonorov, Yu. G., On a class of homogeneous compact Einstein manifolds. (Russian) Sibirsk. Mat. Zh. 41(2000), no. 1, 200–205; Engl. transl. in Siberian Math. J. 41(2000), no. 1, 168–172. Google Scholar

[18] [18] Sakane, Y., Homogeneous Einstein metrics on flag manifolds. Lobachevskii J. Math. 4(1999), 71–87. Google Scholar

[19] [19] N., Wallach: Compact homogeneous riemannian manifolds with strictly positive curvature, Ann. math. 96 (1972) 277–295. Google Scholar

[20] [20] Wang, M., Einstein metrics from symmetry and Bundle constructions. In: Surveys in Differential Geometry: Essays on Einstein Manifolds. Surv. Differ. Geom. VI, Int. Press, Boston, MA, 1999, pp. 287–325. Google Scholar

[21] [21] Wang, M.and Ziller, W., Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84(1986), no. 1, 177–194. Google Scholar

[22] [22] Wang, M., Einstein metrics with positive scalar curvature. In: Curvature and Topology of Riemannian Manifolds, Springer Lecture Notes in Mathematics 1201, Springer, Berlin, 1986, pp. 319–336. Google Scholar