Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year


IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 18, Issue 4

Issues

New homogeneous Einstein metrics on quaternionic Stiefel manifolds

Andreas Arvanitoyeorgos / Yusuke Sakane
  • Osaka University, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Suita, Osaka 565-0871, Japan
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Marina Statha
Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0014

Abstract

We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(np) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpn ≅ Sp(n)/Sp(np), where n = k1 + k2 + k3 and p = nk3. We view Vpn 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(np)).

Keywords: Homogeneous space; Einstein metric; quaternionic Stiefel manifold; generalized Wallach space; generalized flag manifold; isotropy representation, Gröbner basis

MSC 2010: Primary 53C25; Secondary 53C30; 13P10; 65H10; 68W30

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 1165.53342Google 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 1183.53037Google 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 1253.53043Google Scholar

  • [4]

    A. Arvanitoyeorgos, Y. Sakane, M. Statha, New homogeneous Einstein metrics on Stiefel manifolds. Differential Geom. Appl. 35 (2014), 2–18. MR3254287 Zbl 1327.53055Google 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 1333.53053Google Scholar

  • [6]

    A. L. Besse, Einstein manifolds. Springer 1987. MR867684 Zbl 0613.53001Google 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 1068.53029Google Scholar

  • [8]

    G. R. Jensen, Einstein metrics on principal fibre bundles. J. Differential Geometry 8 (1973), 599–614. MR0353209 Zbl 0284.53038Google Scholar

  • [9]

    J.-S. Park, Y. Sakane, Invariant Einstein metrics on certain homogeneous spaces. Tokyo J. Math. 20 (1997), 51–61. MR1451858 Zbl 0884.53039Google Scholar

  • [10]

    M. Statha, Invariant metrics on homogeneous spaces with equivalent isotropy summands. Toyama Math. J. 38 (2016), 35–60. MR3675273 Zbl 1371.53052Google 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 1003.53037Google Scholar

  • [12]

    M. Y. Wang, W. Ziller, Existence and nonexistence of homogeneous Einstein metrics. Invent. Math. 84 (1986), 177–194. MR830044 Zbl 0596.53040Google 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 1262.53044Google Scholar

  • [14]

    W. Ziller, Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259 (1982), 351–358. MR661203 Zbl 0469.53043Google Scholar

About the article


Received: 2016-09-05

Published Online: 2018-10-25

Published in Print: 2018-10-25


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.


Citation Information: Advances in Geometry, Volume 18, Issue 4, Pages 509–524, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0014.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Comments (0)

Please log in or register to comment.
Log in