Relations between space-structures and curvatures

这个研究项目的首席研究员T.Sakai一直致力于这一研究主题：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。在目前的科学研究资助计划的支持下，他特别研究了黎曼流形中距离函数的行为。他开始研究黎曼流形的结构，其中包含一个函数f，其梯度是常范数的，项目标题为“空间的曲率和结构”，由科学研究补助金(C)(2)，编号09640109(1997年至1998年)资助。这是距离函数的显著性质之一。他利用f的拉普拉斯函数得到了模型翘曲乘积情形的等价情形的刻画，并研究了结果的扰动形式，其中Ricci曲率起了重要作用。在目前助学金的支持下，他受聘于t…更多的是这项调查的最后一步。关于黎曼流形上距离函数的Morse理论：虽然紧致黎曼流形上距离点p的距离函数f允许f不可微的点，但已知临界点的概念可以像通常的Morse理论那样被引入。然而，距离函数的临界点指数的概念并不明确，Sakai和J.Itoh考虑了p的割轨迹C(P)具有良好的非退化结构的情况。在这种情况下，他们证明了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审查。酒井给一位博士生讲授了“亚历山大空间上的度量不变量和结构定理”，并通过考试得到了一些与球面有关的结果。酒井还发表了综述文章《曲率--直到二十世纪，和未来？》，以及《黎曼流形家族的Ricci曲率有界低于及其极限》。其他研究人员的研究结果：清原确定了椭球体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研究了包括黎曼流形、黎曼多面体和次黎曼流形在内的正则Dirichlet空间的谱收敛。Shioya研究了黎曼流形和拉普拉斯谱的收敛和崩溃。他还致力于亚历山大·桑德罗夫空间的几何和分析。Tamura，主要从解析的角度研究了薛定谔算子和狄拉克算子。较少

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