Ideal boundary of a Hadamard manifold and Kaehler magnetic fields

2016 Fiscal Year Final Research Report

• Project/Area Number: 24540075
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Nagoya Institute of Technology
• Principal Investigator: ADACHI Toshiaki 名古屋工業大学, 工学(系)研究科(研究院), 教授 (60191855)
• Co-Investigator(Renkei-kenkyūsha): OHTSUKA Fumiko 茨城大学, 理学部, 准教授 (90194208) ; MAEDA Sadahiro 佐賀大学, 大学院工学系研究科, 教授 (40181581)
• Research Collaborator: BAO Tuya 内モンゴル民族大学, 数学学院, 准教授
• Project Period (FY): 2012-04-01 – 2017-03-31
• Keywords: Kaehler magnetic fields / Hadamard manifolds / trajectory-harps / ideal boundaries

Outline of Final Research Achievements

A trajectory-harp consists of a trajectory and a variation of geodesics. We first studied string-lengths, lengths of geodesic segments, and zenith angles which are formed by initial vectors of geodesics. Under the condition that sectional curvatures of the underlying manifold are bounded from above, we gave estimates of these quantities from below. We next studied trajectories on a Hadamard Kaehler manifold by using these estimates. When the square of the strength of a Kaehler magnetic field is smaller than the uuper bound of sectional curvatures, we found that every its trajectory is unbounded and that every magnetic exponential map is a diffeomorphism. Moreover, studying trajectory-horns which consist of geodesics and variations of trajectories, we showed that for given a point on a manifold and a point in its ideal boundary there exist a unique trajectory joining them, and that for given distinct points in the ideal boundary there is a trajectory joining them.

Free Research Field

幾何学