Advanced study of integrable geodesic flows

• Project/Area Number: 26400071
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Okayama University
• Principal Investigator: Kiyohara Kazuyoshi 岡山大学, 自然科学研究科, 教授 (80153245)
• Co-Investigator(Kenkyū-buntansha): 伊藤 仁一 椙山女学園大学, 教育学部, 教授 (20193493)
• Project Period (FY): 2014-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 測地線 / 可積分測地流 / リウヴィル多様体 / c-projective equivalence / conjugate locus / 測地線論 / Kaehler-Liouville 多様体 / Liouvill-Staeckel 計量 / Staeckel 系 / 第一積分 / 完全積分可能 / 共役跡 / ラグランジュ特異点 / 楕円体 / thread construction

Outline of Final Research Achievements

We established a certain development for some problems on "integrable geodesic flows". First, we succeeded enlarging the structure theorem on c-projective equivalence problem to certain degenerate case. Second, we studied the conjugate locus on ellipsoid and Liouville manifolds; we specified the form of singularities arising at branch points and clarified their total shape. Third, we developed the classical result of Staude on the drawing of ellipsoid by pins and thread to Liouville manifolds of any dimension. Fourth, we obtained a Kaehlerian analogue of Eisenhart's theorem concerning the integrability of geodesic flows.

Research Products

• [Journal Article] Classification of regular complex 2-polyhedra of positive constant excess (2017)
  • Author(s): Jin-ichi Itoh, Fumiko Ohtsuka
  • Journal Title: Topology and its applications Volume: 219 Pages: 43-54
  • DOI: 10.1016/j.topol.2017.01.004
  • Peer Reviewed / Open Access
• [Journal Article] Thread construction revisited (2016)
  • Author(s): J. Itoh, K. Kiyohara
  • Journal Title: J. Math. Soc. Japan Volume: 68 Pages: 917-938
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Every graph is a cut locus (2015)
  • Author(s): Itoh, Jin-ichi; Vilcu, Costin
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 67 Issue: 3 Pages: 1227-1238
  • DOI: 10.2969/jmsj/06731227
  • NAID: 130005094189
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Costin Moderate smoothness of most Alexandrov surfaces (2015)
  • Author(s): Itoh, Jin-ichi; Rouyer, Joel; Vilcu, Costin
  • Journal Title: Internat. J. Math. Volume: 26 Issue: 04 Pages: 1540004-1540004
  • DOI: 10.1142/s0129167x15400042
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] C-projective equivalence and integrability of the geodesic flow (2015)
  • Author(s): K. Kiyohara
  • Journal Title: Journal of Geometry and Physics Volume: 87 Pages: 286-295
  • Peer Reviewed
• [Presentation] A Kaehlerian analogue of a theorem of Levi-Civita (2017)
  • Author(s): K. Kiyohara
  • Organizer: 測地線と関連する諸問題
  • Place of Presentation: 熊本大学教育学部
  • Year and Date: 2017-01-07
  • Invited
• [Presentation] 測地流の可積分性に関するEisenhartの一定理のケーラー版 (2017)
  • Author(s): 清原一吉
  • Organizer: 神戸研究集会
  • Invited
• [Presentation] 測地流の可積分性に関するEisenhartの定理のケーラー版 (2017)
  • Author(s): 清原一吉
  • Organizer: 福岡大研究集会
  • Invited
• [Presentation] The cut locus of Liouville surfaces and manifolds (2016)
  • Author(s): K. Kiyohara
  • Organizer: The cut licus
  • Place of Presentation: Bangkok, King Mongkut's Institute of Technology
  • Year and Date: 2016-08-03
  • Int'l Joint Research / Invited
• [Presentation] The cut locus of Liouville surfaces and manifolds (2016)
  • Author(s): Kazuyoshi Kiyohara
  • Organizer: The Cut Locus
  • Place of Presentation: King Mongkut's Institute of Technology
  • Year and Date: 2016-08-03
  • Invited
• [Presentation] 楕円体の共役跡の特異点 (2016)
  • Author(s): 清原一吉
  • Organizer: 沼津研究会
  • Place of Presentation: 沼津高専
  • Year and Date: 2016-03-07
  • Invited
• [Presentation] リウヴィル多様体上のヤコビ場について (2015)
  • Author(s): 清原一吉
  • Organizer: 測地線および関連する諸問題
  • Place of Presentation: 熊本大学
  • Year and Date: 2015-01-10
• [Presentation] 楕円体上の共役跡に現れる特異点について (2014)
  • Author(s): 清原一吉
  • Organizer: 広島幾何学研究会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-10-09
• [Presentation] 楕円体上の共役跡とD4+ラグランジュ特異点 (2014)
  • Author(s): 清原一吉
  • Organizer: 日本数学会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-25