Global Studies on Curvature and Geometric Structures of Riemannian Manifolds

黎曼流形曲率和几何结构的全局研究

• 批准号: 13640093
• 负责人: ITOKAWA Yoe
• 金额: $ 1.34万
• 依托单位: Fukuoka Institute of Technology
• 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 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640093/
• 关键词: riemannian manifolds; sectional curvature; comparison theorems; locally pseudoconvex spaces; extension of functions; Hahn-Banach type theorems; Euler eq; Galerkin approximation; 最短閉側地線; 局所擬凸複素空間; 正則関数の延長; 局所凸空間; トポノゴフの三角形比較定理

Head investigator, Itokawa in the papers "Maximal diameter theorem for manifolds with restricted radial curvature" and "Generalized Toponogov's theorem for manifolds with radial curvature bounded below" written jointly with Katsuhiro Shiohama of Saga University and Yoshiro Machigashira of Osaka University of Educations, investigated riemannian manifolds whose sectional curvature in radial directions from a fixed point is bounded from below by certain function. We have obtained some triangle comparison theorems generalizing that of Toponogov. Moreover, in case equality is attained in these comparisons, these manifolds exhibit very rigid geometrical structure containing some totally geodesic surfaces. We have also obtained some applications of these results including a new sphere theorem.Investigator Nishihara is continuing his study on the extendibility of certain functions on infinite dimensional locally pseudo-convex spaces. In "The extension of polynomials of integral type in locally … More convex spaces and its applications" and "The extension of entire functions of nuclear type on locally convex spaces", he has extended his own results on the extendibility to the total space of polynomials and holomorphic functions which are a priori only defined on locally convex spaces. In the paper "The extension of entire function of nuclear type", he has also succeeded in generalizing Hahn-Banach type theorems of Meise-Vogt and of Nishihara himself. Further extension will be discussed in a forthcoming paper "A Hahn-Banach extension theorem for entire functions of nuclear type". In another forthcoming paper "Pseudoconvex domains of infinite dimensional Grassmann manifolds", Nishihara has also proved a vanishing theorem for regular pseudoconvex domains in Banach spaces.Nishiyama who has substituted as an investigator for the year 2001 has written two papers "Pseudo-advection methods for the axisymmetric stationary Euler equations" and "Magnetohydrodynamic approach to the solvability of the three-dimensional stationary Euler equations". In these papers, Nishiyama studied the methods for obtaining stationary solutions to certain Euler type equations. In particular, he established the effectiveness of the method of using the Galerkin approximation on certain accompanying equations associated to the given equation in case the given equation has a rotationary symmetry around the axis and for general equations in dimension 3. Less

首席研究员，Itokawa在论文“最大直径定理的流形与限制径向曲率”和“广义Toponogov定理的流形与径向曲率有界以下”共同撰写的佐贺大学的盐滨克弘和大坂教育大学的町头义郎，研究了黎曼流形的截面曲率在径向方向从一个固定点是有界的从下面的某些功能。我们得到了一些三角比较定理，推广了Toponogov的定理.此外，在平等的情况下，在这些比较中，这些流形表现出非常刚性的几何结构，包含一些全测地表面。我们也得到了这些结果的一些应用，包括一个新的领域theorem.Investigator西原是继续他的研究在无限维局部伪凸空间的某些功能的可拓性。在“局部积分型多项式的扩张”中， ...更多信息 凸空间及其应用”和“的扩展整个职能的核型局部凸空间”，他已扩大了自己的结果的可扩展性的总空间的多项式和全纯函数是先验只定义在局部凸空间。在论文“延伸的整个功能的核型”，他还成功地推广哈恩-巴拿赫型定理的梅泽-福格特和西原本人。进一步的扩展将在即将发表的论文“A Hahn-Banach extension theorem for entire functions of nuclear type”中讨论。在另一篇即将发表的论文“无限维格拉斯曼流形的伪凸域”中，西原还证明了消失定理经常pseudoconvular域在Banach空间。西山谁已取代作为一个调查员为2001年写了两篇论文“伪平流方法的轴对称静止欧拉方程”和“磁流体动力学方法的可解性的三维静止欧拉方程”。在这些论文中，西山研究了获得某些欧拉型方程定常解的方法。特别是，他建立了有效的方法，使用伽辽金近似的某些附带方程相关的给定方程的情况下，给定的方程有一个旋转对称的轴和一般方程在3维。少

项目成果

Takahiro Nishiyama: "Magnetohydrodynamic approach to solvability of the three-dimensional stationary Euler equations"Glasgow Mathematical Journal. (to appear).

Takahiro Nishiyama：“三维稳态欧拉方程可解性的磁流体动力学方法”格拉斯哥数学杂志。

Yoe Itokawa, Yoshiro Machibashira, Kathuhiro Shiohama: "Maximal diameter theorems for manifolds with restricted radial curvature"Proceedings of the Fifth Pacific Rim Geometry Conference. 61-68 (2001)

Yoe Itokawa、Yoshiro Machibashira、Kathuhiro Shiohama：“具有受限径向曲率的流形的最大直径定理”第五届环太平洋几何会议论文集。

Masaru Nishihara: "The extension of entire functions of nuclear type on locally convex spaces"Proceedings of the Ninth International Colloquium on Finite or Infinite Dimensional Complex Analysis. (発表予定).

Masaru Nishihara：“核类型的整个函数在局部凸空间上的扩展”第九届有限或无限维复分析国际学术研讨会论文集（待提交）。

Masaru Nishihara: "The extension of polynomials of integral type in locally convex spaces and its applications"Proceedings of the Eighth International Colloquium on Finite or Infinite Dimensional Complex Analysis. 161-165 (2001)

Masaru Nishihara：“局部凸空间中积分型多项式的推广及其应用”第八届国际有限或无限维复分析学术研讨会论文集。

Takahiro Hishiyama: "Magnetohydrodynamic approach to solvability of the three-dimensional stationary Euler equations"Glasgow Mathematical Journal. (発表予定).

Takahiro Hishiyama：“三维稳态欧拉方程可解性的磁流体动力学方法”格拉斯哥数学杂志（待出版）。