Differential geometry and integral geometry in homogeneous spaces and its applications

• Project/Area Number: 21540063
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: University of Tsukuba
• Principal Investigator: TASAKI Hiroyuki 筑波大学, 数理物質系, 准教授 (30179684)
• Co-Investigator(Renkei-kenkyūsha): ITOH Mitsuhiro 筑波大学, 名誉教授 (40015912) ; IKAWA Osamu 福島工業高等専門学校, 一般教科, 教授 (60249745) ; SAKAI Takashi 首都大学東京, 理工学研究科, 准教授 (30381445)
• Research Collaborator: TANAKA Makiko 東京理科大学, 理工学部, 准教授 (20255623) ; IRIYEH Hiroshi 東京電機大学, 未来科学部, 准教授 (30385489)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2011: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2010: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2009: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 微分幾何学 / 積分幾何学 / 等質空間 / 部分多様体 / Hermite対称空間 / 実形 / 対蹠集合 / 積分幾河学 / Hermite対称空問 / 交叉

Research Abstract

We proved that the intersection of two real forms in a Hermitian symmetric space of compact type is an antipodal set in a joint work with Makiko Tanaka and investigated their intersection by the use of the theory of polars introduced by Chen-Nagano. Using this result we determined the Floer homology of two real forms and extended Arnold-Givental inequalities in a joint work with Hiroshi Iriyeh and Takashi Sakai. Moreover we used these results and a kinematic formula to obtain an estimate of the volume of Hamiltonian deformation of a real form from below.

Research Products

• [Journal Article] The intersection of two realforms in the complex hyperquadric (2010)
  • Author(s): H. Tasaki
  • Journal Title: TohokuMath. J Volume: 62 Pages: 375-382
  • NAID: 130004713650
  • Peer Reviewed
• [Journal Article] Lie-algebraic characterization oftangentially degenerate orbits ofs-representations (2010)
  • Author(s): O. Ikawa, T. Sakai and H. Tasaki
  • Journal Title: Differential Geom.Appl Volume: 28 Pages: 81-101
  • Peer Reviewed
• [Journal Article] Lie- algebraic characterization of tangentially degenerate orbits of s-representations (2010)
  • Author(s): O.Ikawa, T.Sakai, H.Tasaki
  • Journal Title: Differentail Geometry and its Applications Volume: 28 Pages: 81-101
  • Peer Reviewed
• [Journal Article] The intersection of two real forms in the complex hyperquadric (2010)
  • Author(s): H.Tasaki
  • Journal Title: Tohoku Mathematical Journal Volume: 62 Pages: 375-382
  • NAID: 130004713650
  • Peer Reviewed
• [Journal Article] Lie- algebraic characterization of tangentially degenerate orbits of s-representations (2010)
  • Author(s): O.Ikawa, T.Sakai, H.Tasaki
  • Journal Title: Differentail Geometry and its Applications 28 Pages: 81-101
  • Peer Reviewed
• [Journal Article] Convergence rates ofapproximate sums of Riemann integrals (2009)
  • Author(s): H. Tasaki
  • Journal Title: J.Approx. Theory Volume: 161 Pages: 477-490
  • Peer Reviewed
• [Journal Article] Weaklyreflective submanifolds and austeresubmanifolds (2009)
  • Author(s): O. Ikawa, T. Sakai and H. Tasaki
  • Journal Title: J. Math. Soc. Japan Volume: 61 Pages: 437-481
  • Peer Reviewed
• [Journal Article] Weakly reflective submanifolds and austere submanifolds (2009)
  • Author(s): O.Ikawa, T.Sakai, H.Tasaki
  • Journal Title: Journal of Mathematical Society of Japan 61 Pages: 437-481
  • NAID: 10024905762
  • Peer Reviewed
• [Journal Article] Convergence rates of approximate sums of Riemann integrals (2009)
  • Author(s): H.Tasaki
  • Journal Title: Journal of Approximation Theory 161 Pages: 477-490
  • NAID: 120007131246
  • Peer Reviewed
• [Journal Article] Lagrangian Floer homology of a pair of real forms in Hermitian symmetric spaces of compact type
  • Author(s): H. Iriyeh, T. Sakai and H. Tasaki
  • Journal Title: J. Math. Soc. Japan
  • NAID: 130003384667
  • Peer Reviewed
• [Journal Article] Antipodalsets of symmetric R-spaces
  • Author(s): M. S. Tanaka and H. Tasaki
  • Journal Title: Osaka J. Math
  • Peer Reviewed
• [Journal Article] The intersection of two real forms in Hermitian symmetric spaces of compact type
  • Author(s): M. S. Tanaka and H. Tasaki
  • Journal Title: J. Math. Soc. Japan
  • NAID: 10031177268
  • Peer Reviewed
• [Presentation] Lagrangian Floer homologyand its application to Hamiltonian volumeminimizing property (2012)
  • Author(s): 酒井高司
  • Organizer: The 4th TIMS-OCAMIJoint International Workshop on Differential Geometry and Geometric Analysis
  • Place of Presentation: National Taiwan University, Taiwan
  • Year and Date: 2012-03-18
• [Presentation] Volume estimates of Lagrangian submanifolds in complex hyperquadrics (2011)
  • Author(s): 田崎博之
  • Organizer: 2011 International conference on Integral and Convex Geometric Analysis and Related Topics
  • Place of Presentation: Chern Institute of Mathematics, Nankai University, China
  • Year and Date: 2011-12-16
• [Presentation] 複素二次超曲面の実形のHamilton体積最小性について (2011)
  • Author(s): 酒井高司
  • Organizer: 部分多様体論・湯沢2011
  • Place of Presentation: 湯沢グランドホテル
  • Year and Date: 2011-11-25
• [Presentation] 対蹠集合とその応用 (2011)
  • Author(s): 田崎博之
  • Organizer: 研究集会「部分多様体幾何とリー群作用」
  • Place of Presentation: 東京理科大学森戸記念館
  • Year and Date: 2011-09-03
• [Presentation] 対蹠集合とその応用 (2011)
  • Author(s): 田崎博之
  • Organizer: 研究集会「部分多様体幾何とりー群作用」
  • Place of Presentation: 東京理科大学森戸記念館(東京都)(招待講演)
  • Year and Date: 2011-09-03
• [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉 (2011)
  • Author(s): 田崎博之
  • Organizer: 研究集会:有界対称領域の算術商のコホモロジーとモジュラー・サイクル
  • Place of Presentation: 東京大学玉原国際セミナーハウス
  • Year and Date: 2011-08-04
• [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉 (2011)
  • Author(s): 田崎博之
  • Organizer: 研究集会:有界対称領域の算術商のコホモロジーとモジュラー・サイクル
  • Place of Presentation: 東京大学玉原国際セミナーハウス(群馬県)(招待講演)
  • Year and Date: 2011-08-04
• [Presentation] コンパクト対称空間の幾何学 (2010)
  • Author(s): 田中真紀子, 田崎博之
  • Organizer: 秋葉原微分幾何セミナー
  • Place of Presentation: 秋葉原ダイビル
  • Year and Date: 2010-12-04
• [Presentation] コンパクト対称空間の対蹠集合 (2010)
  • Author(s): 田崎博之
  • Organizer: 幾何学阿蘇研究集会
  • Place of Presentation: 休暇村南阿蘇
  • Year and Date: 2010-09-01
• [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉 (2010)
  • Author(s): 田崎博之
  • Organizer: 第57回幾何学シンポジウム
  • Place of Presentation: 神戸大学
  • Year and Date: 2010-08-08
• [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉 (2010)
  • Author(s): 田崎博之
  • Organizer: 第57回幾何学シンポジウム
  • Place of Presentation: 神戸大学 招待講演
  • Year and Date: 2010-08-08
• [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉 (2010)
  • Author(s): 田崎博之
  • Organizer: 首都大学東京幾何学セミナー
  • Place of Presentation: 首都大学東京
  • Year and Date: 2010-04-06
• [Presentation] The intersection of two realforms in Hermitian symmetric spaces of compact type (2010)
  • Author(s): 田崎博之
  • Organizer: Mini-Workshop : Valuationsand Integral Geometry
  • Place of Presentation: Mathematisches Forschungsinstitut Oberwolfach, Germany
  • Year and Date: 2010-01-21
• [Presentation] The intersection of two real forms in Hermitian symmetric spaces of compact type (2010)
  • Author(s): 田崎博之
  • Organizer: Mini-Workshop : Valuations and Integral Geometry
  • Place of Presentation: Oberwolfach数学研究所(ドイツ)
  • Year and Date: 2010-01-21
• [Presentation] コンパクト型Hermite対称空間の二つの実形の交叉 (2009)
  • Author(s): 田崎博之
  • Organizer: 部分多様体論・湯沢2009
  • Place of Presentation: 湯沢グランドホテル
  • Year and Date: 2009-11-26
• [Presentation] 複素Grassmann多様体の二つの実形の交叉 (2009)
  • Author(s): 田崎博之
  • Organizer: 広島大学幾何学研究集会
  • Place of Presentation: 広島大学
  • Year and Date: 2009-10-07
• [Presentation] The intersection of two realforms in the complex hyperquadric (2009)
  • Author(s): 田崎博之
  • Organizer: 研究集会「部分多様体幾何とリー群作用」
  • Place of Presentation: 東京理科大学森戸記念館
  • Year and Date: 2009-09-07
• [Remarks]
  • URL: http://www.tsukuba.ac.jp/~tasaki/
• [Remarks]
  • URL: http://www.math.tsukuba.ac.jp/~tasaki/
• [Remarks]
  • URL: http://www.math.tsukuba.ac.jp/~tasaki/
• [Remarks]
  • URL: http://www.math.tsukuba.ac.jp/~tasaki/