Study of integrable geodesic flows and related problems

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K03302
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Okayama University
• Principal Investigator: Kiyohara Kazuyoshi 岡山大学, 自然科学研究科, 特命教授 (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): 2018-04-01 – 2021-03-31
• Keywords: 可積分測地流 / ハミルトン力学 / 共役跡 / ラグランジュ特異点 / 射影同値 / C射影同値

Outline of Final Research Achievements

A paper with J. Itoh, which was one of the basis of this research, was published on Arnold Math. J., 31-90 (2001) . One of our targets of this research was to extend it to non compact case, and we established the first step by constructing projective embeddings of hyper quadrics into products of spheres.
Next, we studied the PQ equivalence of Topalov on Hermite-Liouville manifolds and gave a generalization of his "PQ-hierarchy". Moreover, we showed that a local inverse of this statement is correct, which would indicate that the notion of PQ equivalence is not so far from that of "C projective equivalence".

Free Research Field

微分幾何学

Academic Significance and Societal Importance of the Research Achievements

可積分測地流はそれ独自への関心とともに、特異点論、射影同値など他の観点からも興味深い対象となって来ている。本研究は特に共役跡に現れる特異点の問題と、リーマン多様体の射影同値、ケーラー多様体のC射影同値、さらにはエルミート多様体のPQ射影同値に至る一連の概念について、独自の方向を示し、新たな進展をもたらすものである。