Studies on diameter conjecture on flat tori in the unit sphere

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04836
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Utsunomiya University
• Principal Investigator: Kitagawa Yoshihisa 宇都宮大学, 教育学部, 教授 (20144917)
• Co-Investigator(Renkei-kenkyūsha): AIHARA Yosihiro 福島大学, 人間発達文化学類, 教授 (60175718) ; UMEHARA Masaaki 東京工業大学, 大学院情報理工学研究科, 教授 (90193945)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: 微分幾何 / 部分多様体 / 平坦トーラス / ３次元球面 / 直径 / 剛性 / 正則閉曲線 / ２重接触

Outline of Final Research Achievements

Diameter conjecture on flat tori in the unit 3-sphere states that the extrinsic diameter of isometrically immersed flat tori in the unit 3-sphere is equal to π. To prove this conjecture, it is sufficient to prove bi-tangent conjecture on periodic admissible pairs (c_1,c_2) in the unit 2-sphere, which states that if the self-intersection numbers of the closed curves c_1 and c_2 are odd, then c_1 and c_2 have a bi-tangent of the second kind．In this research, we study bi-tangent conjecture on periodic admissible pairs (c_1,c_2), and we proved that the conjecture is true if the curve c_1 contains an negative shell.

Free Research Field

数学（幾何学）