Studies on some open problems concerning flat tori in odd dimensional spheres

2014 Fiscal Year Final Research Report

• Project/Area Number: 24540066
• 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 Yoshihiro 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): 2012-04-01 – 2015-03-31
• Keywords: 微分幾何 / 部分多様体 / 平坦トーラス / ３次元球面 / 直径予想 / 剛性定理 / 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. In 2011, we obtained a theorem which states that the conjecture is true under the assumption that the mean curvature of the immersion is nonnegative or nonpositive. Unfortunately, there was a gap in the proof of a proposition which was a key assertion to prove the theorem. In this research, using a new lemma on the shape of simple loops in the unit 2-sphere, we corrected the proof of the proposition. As a result, we established the theorem.

Free Research Field

数学（幾何学）