Relations between space-structures and curvatures

该研究项目的首席研究员酒井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及其与流形结构的联系。在本年度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。在国家科学研究资助项目(C) (2), no . 09640109（1997-1998）的资助下，开始研究承认梯度为常范数函数f的黎曼流形的结构。这是距离函数的一个重要性质。他将模型翘曲积的情况描述为f的拉普拉斯不等式的相等情况，并研究了结果的扰动版本，其中里奇曲率起了重要作用。在本赠款的支持下，他正在从事这项调查的最后一步——……More。黎曼流形上距离函数的莫尔斯理论：虽然距离函数f到紧黎曼流形M上点p的距离函数f允许f不可导的点，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标的概念并不明确，Sakai与J. Itoh一起考虑了p的切割轨迹C(p)具有良好的非简并结构的情况。在这种情况下，他们证明了切割轨迹允许惠特尼分层，并发展了距离函数的莫尔斯理论，引入了临界点指数的概念。另一方面，后来发现有V. Gerschkovich和H. Rubinstein的相关著作，我们需要对这个问题进行更多的研究。Sakai给一名博士生做了一个关于“Alexsandrov空间上的韵律不变量和结构定理”的题目，并通过检验得到了一些与球有关的结果。酒井还参与出版了调查文章《曲率——直到20世纪，以及未来？》和“Ricci曲率有界的黎曼流形族及其极限”。其他研究者的研究成果：清原确定了椭球体中任意点切割轨迹的显式结构。Katsuda研究了Neumann边值问题的逆问题，Kasue研究了正则Dirichlet空间包括黎曼流形、黎曼多面体和次黎曼流形的谱收敛性。Shioya研究了黎曼流形的收敛和坍缩以及拉普拉斯算子的谱。他还积极研究几何和亚历山德罗夫空间的分析。Tamura主要从分析的角度研究了薛定谔算子和狄拉克算子。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