Global studies of principal distributions on surfaces and researches of principal distributions on various submanifolds

2014 Fiscal Year Final Research Report

• Project/Area Number: 24740048
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Kumamoto University
• Principal Investigator: ANDO Naoya 熊本大学, 自然科学研究科, 准教授 (50359965)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Keywords: 主分布 / 過剰決定系 / 整合条件 / 平均曲率ベクトルが零 / 複素曲線 / 正則３次微分 / アファインSchwarz写像 / 球面Schwarz写像

Outline of Final Research Achievements

Referring to results of over-determined systems on surfaces in the Euclidean space, I obtained the corresponding results of over-determined systems on surfaces in the other 3-dimensional space forms. I obtained characterizations of spacelike surfaces with zero mean curvature vector in the 4-dimensional Riemannian and Lorentz space forms in terms of the induced metrics and principal distributions w.r.t. suitable normal vector fields. A minimal surface in the 4-dimensional Euclidean space s.t. principal curvatures do not depend on the choice of a unit normal vector is congruent with a complex curve in the two-dimensional complex number space and characterized in terms of the induced metric and a holomorphic cubic differential. In relation to this characterization, any complex curve is locally considered as the image of the composition of an affine Schwarz map and a parallel translation. I obtained a characterization of a sphere Schwarz immersion in terms of a positive-valued function.

Free Research Field

微分幾何学