Relations between space-structures and curvatures

该研究项目的首席研究员T.Sakai一直致力于研究主题：黎曼流形的各种度量不变量之间的关系，以及它们与流形结构的联系。在目前的科学研究补助金的支持下，他特别研究了黎曼流形中距离函数的行为。他开始研究结构的黎曼流形承认一个函数f的梯度是常数规范下的项目标题“曲率和结构的空间”支持的赠款援助科学研究（C）（2），编号。09640109（1997-1998年）。这是距离函数的一个显著性质。他得到的表征模型翘曲产品的情况下平等的情况下不等式的拉普拉斯算子的f，并研究了扰动版本的结果，其中里奇曲率发挥了重要作用。在现有的补助金的支持下，他从事了 关于我们 这是调查的最后一步。黎曼流形上距离函数的莫尔斯理论：虽然紧致黎曼流形M上一个点p的距离函数f包含f不可微的点，但已知临界点的概念可以像通常的莫尔斯理论一样引入。然而，距离函数的临界点指数的概念并不清楚，Sakai和J. Itoh一起考虑了p的切割轨迹C（p）具有良好的非退化结构的情况。他们表明，在这种情况下，削减轨迹承认惠特尼分层和发展莫尔斯理论的距离函数介绍的概念指数的临界点。另一方面，后来发现有V. Gerschkovich和H.鲁宾斯坦，我们需要对这个问题进行更多的研究。酒井作了一个主题“度量不变量和结构定理的Alexsandrov空间”的博士课程的学生，并通过考试的一些结果有关的领域获得。酒井还曾为出版调查文章“曲率--直到二十世纪，和未来？和“具有Ricci曲率下界的黎曼流形族及其极限”.其他研究者的研究成果：清原确定了椭球中任意点的切割轨迹的显式结构。Katsuda研究了Neumann边值问题的反问题，Kasue研究了正则Dirichlet空间的谱收敛，包括黎曼流形，黎曼多面体和子黎曼流形。Shioya研究了黎曼流形的收敛和坍缩以及拉普拉斯算子的谱。他还积极致力于几何和分析Alexsandrov空间。Tamura主要从分析的角度研究了Schroedinger算子和Dirac算子。少

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