New homogeneous Einstein metrics on quaternionic Stiefel manifolds
-
Andreas Arvanitoyeorgos
,
Yusuke Sakane
Abstract
We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).
Funding: The first and third authors were supported by Grant #E.037 from the Research Committee of the University of Patras (Programme K. Karatheodori). The second author was supported by JSPS KAKENHI Grant Number JP16K05130. The first author was supported by a grand from the Empirikion Foundation in Athens, Greece.
References
[1] A. Arvanitoyeorgos, V. V. Dzhepko, Y. G. Nikonorov, Invariant Einstein metrics on quaternionic Stiefel manifolds. Bull. Greek Math. Soc. 53 (2007), 1–14. MR2466490 Zbl 1165.53342Search in Google Scholar
[2] A. Arvanitoyeorgos, V. V. Dzhepko, Y. G. Nikonorov, Invariant Einstein metrics on some homogeneous spaces of classical Lie groups. Canad. J. Math. 61 (2009), 1201–1213. MR2588419 Zbl 1183.5303710.4153/CJM-2009-056-2Search in Google Scholar
[3] A. Arvanitoyeorgos, K. Mori, Y. Sakane, Einstein metrics on compact Lie groups which are not naturally reductive. Geom. Dedicata 160 (2012), 261–285. MR2970054 Zbl 1253.5304310.1007/s10711-011-9681-1Search in Google Scholar
[4] A. Arvanitoyeorgos, Y. Sakane, M. Statha, New homogeneous Einstein metrics on Stiefel manifolds. Differential Geom. Appl. 35 (2014), 2–18. MR3254287 Zbl 1327.5305510.1016/j.difgeo.2014.01.007Search in Google Scholar
[5] A. Arvanitoyeorgos, Y. Sakane, M. Statha, Einstein metrics on the symplectic group which are not naturally reductive. In: Current developments in differential geometry and its related fields, 1–22, World Sci. Publ., Hackensack, NJ 2016. MR3494871 Zbl 1333.5305310.1142/9789814719780_0001Search in Google Scholar
[6] A. L. Besse, Einstein manifolds. Springer 1987. MR867684 Zbl 0613.5300110.1007/978-3-540-74311-8Search in Google Scholar
[7] C. Böhm, M. Wang, W. Ziller, A variational approach for compact homogeneous Einstein manifolds. Geom. Funct. Anal. 14 (2004), 681–733. MR2084976 zbl 1068.5302910.1007/s00039-004-0471-xSearch in Google Scholar
[8] G. R. Jensen, Einstein metrics on principal fibre bundles. J. Differential Geometry 8 (1973), 599–614. MR0353209 Zbl 0284.5303810.4310/jdg/1214431962Search in Google Scholar
[9] J.-S. Park, Y. Sakane, Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math. 20 (1997), 51–61. MR1451858 Zbl 0884.5303910.3836/tjm/1270042398Search in Google Scholar
[10] M. Statha, Invariant metrics on homogeneous spaces with equivalent isotropy summands. Toyama Math. J. 38 (2016), 35–60. MR3675273 Zbl 1371.53052Search in Google Scholar
[11] M. Y. Wang, Einstein metrics from symmetry and bundle constructions. In: Surveys in differential geometry: essays on Einstein manifolds, volume 6 of Surv. Differ. Geom. 287–325, Int. Press, Boston, MA 1999. MR1798614 Zbl 1003.5303710.4310/SDG.2001.v6.n1.a11Search in Google Scholar
[12] M. Y. Wang, W. Ziller, Existence and nonexistence of homogeneous Einstein metrics. Invent. Math. 84 (1986), 177–194. MR830044 Zbl 0596.5304010.1007/BF01388738Search in Google Scholar
[13] M. Y.-K. Wang, Einstein metrics from symmetry and bundle constructions: a sequel. In: Differential geometry, volume 22 of Adv. Lect. Math., 253–309, Int. Press, Somerville, MA 2012. MR3076055 Zbl 1262.5304410.4310/SDG.2001.v6.n1.a11Search in Google Scholar
[14] W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259 (1982), 351–358. MR661203 Zbl 0469.5304310.1007/BF01456947Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- The index of symmetry of three-dimensional Lie groups with a left-invariant metric
- Canonical contact unit cotangent bundle
- On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces
- Regular polyhedra in the 3-torus
- Rational curves on Del Pezzo manifolds
- Commuting matrices and the Hilbert scheme of points on affine spaces
- On rational varieties of small rationality degree
- Skew symmetric logarithms and geodesics on On(ℝ)
- New homogeneous Einstein metrics on quaternionic Stiefel manifolds
Articles in the same Issue
- Frontmatter
- The index of symmetry of three-dimensional Lie groups with a left-invariant metric
- Canonical contact unit cotangent bundle
- On the umbilicity of generalized linear Weingarten hypersurfaces in hyperbolic spaces
- Regular polyhedra in the 3-torus
- Rational curves on Del Pezzo manifolds
- Commuting matrices and the Hilbert scheme of points on affine spaces
- On rational varieties of small rationality degree
- Skew symmetric logarithms and geodesics on On(ℝ)
- New homogeneous Einstein metrics on quaternionic Stiefel manifolds
