Relations between space-structures and curvatures

空间结构与曲率之间的关系

• 批准号: 12440020
• 负责人: SAKAI Takashi
• 金额: $ 4.86万
• 依托单位: OKAYAMA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440020/
• 关键词: Riemannian manifolds; Alexandrov spaces; Curvatures; Metric Invariants; Distance functions

T.Sakai, head investigator of this research program, has been working on the research theme : relationships between various metrical invariants of Riemannian manifolds, and their connection with the manifold structure. Under the support of the present Grant-in-Aid for Scientific Research, he especially studied the behavior of distance functions in Riemannian manifolds.1. He begun to study the structure of Riemannian manifolds admitting a function f whose gradient is of constant norm under the project title "Curvature and structure of spaces" supported by the Grant-in-Aid for Scientific Research (C) (2), Nr. 09640109 (1997-1998). This is one of the remarkable properties of distance functions. He obtained characterizations of model warped product cases as equality case of inequalities in terms of the Laplacian of f, and investigated the perturbed version of the result, where the Ricci curvature played an important role. Under the support of the present Grant-in-Aid, he was engaged with t … More he final step of this investigation.2. Morse theory for a distance function on a Riemannian manifold : Although distance function f from a point p of a compact Riemannian manifold M admits points where f is not differentiable, it was known that the notion of critical points may be introduced as in usual Morse theory. However, the notion of the index of critical points of distance functions was not clear, and Sakai considered with J. Itoh the case where the cut locus C(p) of p carries a nice non-degeneracy structure. They showed in this case that the cut locus admits the Whitney stratification and developed Morse theory for distance function introducing the notion of the index of critical points. On the other hand, it later turned out that there are related works by V. Gerschkovich and H. Rubinstein, and we need more examination on the problem. Sakai gave a theme on "metrical invariants and the structure theorems on Alexsandrov spaces" to a student of doctor course and through examination some results related to the spheres were obtained. Sakai also worked for publication of survey articles "Curvature --Until the twentieth century, and the future? ", and "Family of Riemannian manifolds with Ricci curvature bounded below and its limits".3. Research results of other investigators : Kiyohara determined the explicit structure of the cut locus of any point in ellipsoids. Katsuda studied the inverse problem of the Neumann boundary value problem, and Kasue investigated the spectral convergence of regular Dirichlet spaces including Riemannian manifolds, Riemannian polyhedra and sub-Riemannian manifolds. Shioya studied convergence and collapsing of Riemannian manifolds and spectrum of Laplacians. He also vigorously worked on geometry and analysis of Alexsandrov spaces. Tamura, studied Schroedinger operators and Dirac operators mainly from analytical viewpoint. Less

该研究计划的负责人T.Sakai一直在研究研究主题：Riemannian流形的各种度量不变之间的关系及其与歧管结构的联系。在目前的科学研究赠款的支持下，他特别研究了Riemannian歧管中距离功能的行为1。他开始研究Riemannian歧管的结构，承认该功能F在项目标题“曲率和空间的曲率和结构”下，由科学研究授予的授予（C）（2），NR。 09640109（1997-1998）。这是距离函数的显着特性之一。他获得了模型扭曲的产品案例的特征，作为F的Laplacian的平等案例，并研究了结果的扰动版本，其中RICCI曲率发挥了重要作用。在目前的赠款的支持下，他与T ...更最后一步。莫尔斯（Morse）在riemannian歧管上的距离函数的理论：尽管从紧凑的riemannian歧管M的距离p f距离函数f，其中f无法区分，但众所周知，临界点的概念可以像通常的摩尔斯理论中一样引入。但是，距离距离函数临界点的索引的概念尚不清楚，而sakai则在J. Itoh中考虑了p的切割基因座C（p）带有一个不错的非脱位结构。在这种情况下，他们表明，切割基因座接受了惠特尼分层，并开发了用于距离函数的摩尔斯理论，引入了临界点索引的概念。另一方面，后来发现V. Gerschkovich和H. Rubinstein有相关的作品，我们需要对这个问题进行更多检查。 Sakai向医生课程的学生提供了一个主题，主题为“ Metrical不变性和Alexandrov空间的结构定理”，并通过考试获得了与球体有关的一些结果。 Sakai还在发表调查文章“曲率 - 直到二十世纪以及未来？”和“ Riemannian流形的家族？ricci曲率下面构成了下面的RICCI曲率及其限制”。3。其他研究者的研究结果：Kiyohara确定了椭圆形任何点的切割基因座的明确结构。 Katsuda研究了Neumann边界价值问题的反问题，Kasue研究了包括Riemann歧管，Riemannian Polyhedra和Sub-Riemannian歧管的常规迪里奇空间的光谱收敛性。 Shioya研究了Riemannian歧管和Laplacians谱的融合和崩溃。他还大力研究了亚历山德罗夫空间的几何形状和分析。 Tamura，Studiod Schroedinger操作员和Dirac运营商主要来自分析观点。较少的

项目成果

K.Shiohama, Takashi Shioya, M.Tanaka: "The Geometry of Total Curvature on Complete Open Surfaces"Cambridge Univ.Press (To appear). (2003)

K.Shiohama、Takashi Shioya、M.Tanaka：“完全开放曲面上总曲率的几何”剑桥大学出版社（待出版）。

K.Kuwae, Takashi Shioya: "Sobolev and Dirichlet spaces over maps between metric spaces"J.Reine Angew.Math.. 555. 39-75 (2003)

K.Kuwae、Takashi Shioya：“度量空间之间的映射上的 Sobolev 和 Dirichlet 空间”J.Reine Angew.Math.. 555. 39-75 (2003)

T.Ichinose, Hideo Tamura: "The norm convergence of the Trotter-Kato product formula with error bound"Comm.Math.Phys.(2001). 217. 489-502 (2001)

T.Ichinose、Hideo Tamura：“具有误差界限的 Trotter-Kato 乘积公式的范数收敛性”Comm.Math.Phys.(2001)。

Mitsuhiro Miyazaki and Yuji Yoshino: "On heights and grades of determinatal ideals"Journal of Algebra. 235. 783-804 (2001)

Mitsuhiro Miyazaki 和 Yuji Yoshino：“论决定性理想的高度和等级”代数杂志。

Takashi Sakai: "Warped products, Ricci curvature and distance functions"Tohoku Mathematical Publications (Proceedings of the Fifth Pacific Rim Geometry Conference). 20. 163-172 (2001)

Takashi Sakai：“翘曲积、利玛窦曲率和距离函数”东北数学出版物（第五届环太平洋几何会议论文集）。