Einstein metrics on compact homogeneous spaces

• Project/Area Number: 25400071
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Osaka University
• Principal Investigator: SAKANE Yusuke 大阪大学, その他部局等, 名誉教授 (00089872)
• Research Collaborator: Andreas Arvanitoyeorgos ; Ioannis Chrysikos ; Marina Statha
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2015: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2014: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2013: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 等質アインシュタイン計量 / 一般化された旗多様体 / Stiefel多様体 / コンパクト単純リー群 / 一般化されたWallach空間 / リッチテンソル / グレブナー基底 / 幾何学 / アインシュタイン計量 / コンパクト等質空間 / 一般化されたWallack空間 / 等質アインシュタイン多様体 / コンパクトリー群上の左不変計量 / 例外型のコンパクト単純リー群 / リー群上の左不変計量のなす空間 / Stiefel 多様体 / リーマンサブマーション / 等質ケーラー計量

Outline of Final Research Achievements

We studied invariant Einstein metrics on compact homogeneous spaces. In particular, we found new Einstein metrics on generalized flag manifolds, Stiefel manifolds and compact simple Lie groups.
We determined all generalized flag manifolds with 5 isotropy summands and found all invariant Einstein metrics on these manifolds. Moreover, we constructed new Einstein metrics on Stiefel manifolds and non naturally reductive Einstein metrics on special orthogonal groups SO(n) (n > 6) and symplectic groups Sp(n) ( n > 2).

Research Products

• [Int'l Joint Research] University of Patras(ギリシャ)
• [Int'l Joint Research] Masaryk University(チェコ)
• [Journal Article] New Einstein metrics on the Lie group SO(n) which are not naturally reductive (2015)
  • Author(s): A. Arvanitoyeorgos, Y. Sakane and M. Statha
  • Journal Title: Geometry, Imaging and Computing Volume: 2 Issue: 2 Pages: 77-108
  • DOI: 10.4310/gic.2015.v2.n2.a1
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The classification of homogeneous Einstein metrics on flag manifolds with b2(M)=1 (2014)
  • Author(s): I. Chrysikos and Y. Sakane
  • Journal Title: Bull. Sci. math. Volume: 138 Issue: 6 Pages: 665-692
  • DOI: 10.1016/j.bulsci.2013.11.002
  • Peer Reviewed
• [Journal Article] New homogeneous Einstein metrics on Stiefel manifolds (2014)
  • Author(s): A. Arvanitoyeorgos, Y. Sakane and M. Statha
  • Journal Title: Differential Geom. Appl. Volume: 35 Pages: 2-18
  • DOI: 10.1016/j.difgeo.2014.01.007
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Proving isometry for homogeneous Einstein metrics on flag manifolds by symbolic computation (2013)
  • Author(s): A. Arvanitoyeorgos, I. Chrysikos and Y. Sakane
  • Journal Title: Journal of Symbolic Computation Volume: 55 Pages: 59-71
  • DOI: 10.1016/j.jsc.2013.03.005
  • Peer Reviewed
• [Journal Article] Homogeneous Einstein Metrics on Generalized Flag Manifolds with five isotropy summands (2013)
  • Author(s): A. Arvanitoyeorgos, I. Chrysikos and Y. Sakane
  • Journal Title: International Journal of Mathematics Volume: 24 Issue: 10 Pages: 1-52
  • DOI: 10.1142/s0129167x13500778
  • Peer Reviewed
• [Presentation] Einstein metrics on compact homogeneous spaces (2016)
  • Author(s): 坂根由昌
  • Organizer: The 4th Workshop "Complex Geometry and Lie Groups"
  • Place of Presentation: 奈良女子大学理学部 （ 奈良市北魚屋西町）
  • Year and Date: 2016-03-25
  • Int'l Joint Research / Invited
• [Presentation] Einstein metrics on compact simple Lie groups (2016)
  • Author(s): 坂根由昌
  • Organizer: Lie Theory and Geometry
  • Place of Presentation: Schloss Rauischholzhausen, ドイツ
  • Year and Date: 2016-03-10
  • Int'l Joint Research / Invited
• [Presentation] Einstein metrics on compact simple Lie groups (2015)
  • Author(s): 坂根由昌
  • Organizer: 2015年度松江セミナー
  • Place of Presentation: 島根大学合理工学部 （ 松江市西川津町）
  • Year and Date: 2015-12-01
• [Presentation] コンパクト単純リー群上の左不変なアインシュタイン計量について (2015)
  • Author(s): 坂根由昌
  • Organizer: 第22回沼津研究会 --幾何，数理物理，そして量子論---
  • Place of Presentation: 沼津工業高等専門学校, 沼津 静岡
  • Year and Date: 2015-03-09
  • Invited
• [Presentation] Some open problems on compact homogeneous Einstein manifolds (2014)
  • Author(s): 坂根由昌
  • Organizer: Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach, Germany
  • Year and Date: 2014-09-29
  • Invited
• [Presentation] New homogeneous Einstein metrics on compact simple Lie groups (2014)
  • Author(s): 坂根由昌
  • Organizer: ICDG2014
  • Place of Presentation: Veliko Tarnovo, Bulgaria
  • Year and Date: 2014-09-08
  • Invited
• [Presentation] New homogeneous Einstein metrics on Stiefel manifolds and compact Lie groups SO(n)
  • Author(s): 坂根由昌
  • Organizer: The 12th Conference on Differential Geometry and its Applications
  • Place of Presentation: Masaryk University Brno, Czech Republic
• [Presentation] New homogeneous Einstein metrics on Stiefel manifolds and compact Lie groups SO(n)
  • Author(s): 坂根由昌
  • Organizer: 広島幾何学研究集会 2013
  • Place of Presentation: 広島大学 大学院先端物質科学研究科（東広島市鏡山）
  • Invited
• [Book] Current Developments in Differential Geometry and its Related Fields (2015)
  • Author(s): A. Arvanitoyeorgos, Y. Sakane and M. Statha
  • Total Pages: 256
  • Publisher: World Sci. Publ.
• [Book] Prospects of Differential Geometry and Its Related Fields: Proceedings of the 3rd International Colloquium on Differential Geometry and Its Related Fields (2013)
  • Author(s): A. Arvanitoyeorgos, I. Chrysikos and Y. Sakane
  • Total Pages: 228
  • Publisher: World Scientific Pub Co Inc