Geometric structures and differential equations on filtered manifolds

滤波流形上的几何结构和微分方程

• 批准号: 13640071
• 负责人: MORIMOTO Tohru
• 金额: $ 1.86万
• 依托单位: Nara Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640071/
• 关键词: filtered manifold; nilpotent geometry; nilpotent analysis; subriemannian manifold; Cartan connection; サブリーマン構造; サブリーマン接触多様体; モンジュ・アンペール方程式; 幾何構造; カルノ・カラテオドリー計量

From the viewpoint of nilpotent geometry and analysis, we have been developing general theories on geometric, structures and differential equations on filtered manifolds. More recently, as applications of these theories, we have been carrying detailed studies on various concrete geometric structures. In particular, applying the general theory of Morimoto to subriemannian geometry, we have obtained the following remarkable theorem : There exists a canonical Cartan connection associated with a subriemannian manifold satisfying Hormander condition and having constant first order approximation.We have also determined, up to quotient by discrete groups, the homogeneous subriemannian contact manifolds whose automorphism groups are of maximal dimension. The automorphism groups are also classified into three isomorphic classes.

从幂零几何和分析的角度出发，我们一直在发展关于滤波流形上的几何、结构和微分方程的一般理论。最近，作为这些理论的应用，我们对各种具体的几何结构进行了详细的研究。特别是，将森本的一般理论应用到亚里曼几何中，我们得到了以下显着的定理：满足霍曼德条件并具有常数一阶近似的亚里曼流形存在一个正则嘉当联系。我们还确定了直到离散群的商，其自同构群具有最大维数的齐次亚里曼接触流形。自同构群也分为三个同构类。

项目成果

T.Morimoto et al.edit: "Lie Groups, Geometric Structures and Differential Equations -One Hundred Years after Sophus Lie-"Advanced studies in Pure Mathematics 37, Mathematical Society of Japan. (2002)

T.Morimoto 等人编辑：“李群、几何结构和微分方程 - Sophus Lie 之后一百年 -”纯数学高级研究 37，日本数学会。

Y.Ueno, Y.Agaoka: "Classification of tilings of the 2-dimensional sphere by congruent triangles"Hiroshima Math. J.. 32. 463-540 (2002)

Y.Ueno，Y.Agaoka：“通过全等三角形对二维球体的平铺进行分类”广岛数学。

H.Sato: "Contact geometry of second order partial differential equations"Sugaku Expositions. (to appear).

H.Sato：《二阶偏微分方程的接触几何》Sugaku Expositions。

H.Sato: "Contact geometry of second order partial differential equations"Sugaku Expositions. (to appear.).

H.Sato：《二阶偏微分方程的接触几何》Sugaku Expositions。

Y.Agaoka, E.Kaneda: "Slrongly ordhogonal subsets in root systems"Hokkaido Math. J.. 31. 107-136 (2002)

Y.Agaoka，E.Kaneda：“根系统中的隆正交子集”北海道数学。