2014 Fiscal Year Final Research Report
Studies on some open problems concerning flat tori in odd dimensional spheres
Project/Area Number 
24540066

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Multiyear Fund 
Section  一般 
Research Field 
Geometry

Research Institution  Utsunomiya University 
Principal Investigator 

CoInvestigator(Renkeikenkyūsha) 
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) 
20120401 – 20150331

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 3sphere states that the extrinsic diameter of isometrically immersed flat tori in the unit 3sphere 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 2sphere, we corrected the proof of the proposition. As a result, we established the theorem.

Free Research Field 
数学（幾何学）
