Metric invariants and space structures

度量不变量和空间结构

• 批准号: 17540079
• 负责人: SAKAI Takashi
• 金额: $ 2.37万
• 依托单位: Okayama University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540079/
• 关键词: Riemannian manifolds; Distance functions; Morse theory; Geodesics; Cut loci; Riemannian manifolds; Distance functions; Morse theory; Geodesics; Cut loci

Distance function d_p to a point p of a compact Riemannian manifold is an important geometric function that is also related to the manifold structure. However, distance function d_p admits a point where d_p is not differentiable, and such a point is contained in the cut locus C_p of p. It was known that the notion of critical points may be introduced as in usual Morse theory and d_p is not differentiable at critical points. In the case where there are no critical points, Morse theory (the isotopy lemma) has been developed as in smooth case that played an important role in applications.Now to develop Morse theory for distance functions including critical points, one should first define the notion of the index at a critical point. Under the support of the present Grant-in-Aid, T. Sakai, head investigator of this research program, carried out this with J. Itoh when Riemannian metric satisfies some natural non-degeneracy conditions, and developed Morse theory for distance functions from a … More direct geometric view point (has been published in a Journal). There it is important to show that the cut locus Cp carries a nice structure (Whitney stratification), and the index of a critical point q is defined in terms of the number of minimal geodesics joining p, q and the usual index at q of the restriction of 4 to the stratum containing q that is a smooth function with a critical point q. I hope to continue to study how generic is the conditions among all Riemannian metrics on a given manifold.With respect to distance functions, K. Kiyohara determined the explicit structure of the cut locus and the conjugate locus of any point in ellipsoids (and some Liouville manifolds) from a view point of integrable geodesic flow with J. Itoh, and A. Katsuda studied the inverse problem of the Neumann boundary value problem, namely how to reconstruct the inner Riemannian metric from the distance function to the boundary of a Riemannian manifold with boundary.As for geometric inequalities (isosystolic inequality, isodiametric inequality) and the first eigenvalue estimate of the Laplacian of a compact Riemannian manifolds with non-negative Ricci curvature and its perturbation, we could not make substantial progress in this period, but Kiyohara obtained a result concerning an inequality of geometric invariants of Alexandrov spaces. I hope to continue to carry the program. Y. Mori investigated algorithm of quantum computer and supported Sakai in computer aid. Less

距离功能D_P到紧凑的Riemannian歧管的点P是一个重要的几何函数，也与歧管结构有关。但是，距离函数D_P承认D_P无法区分的点，并且在p的切割基因座C_P中包含一个点。众所周知，临界点的概念可以像通常的莫尔斯理论中一样引入，并且在临界点上d_p并不可区分。在没有临界点的情况下，莫尔斯理论（同位论引理）是在平滑的情况下在应用中发挥重要作用的。现在，为包括关键点在内的距离函数开发莫尔斯理论，首先应该在关键点上定义索引的概念。在目前的赠款的支持下，该研究计划的首席研究员T. Sakai与J. Itoh进行了此操作，当Riemannian Metric满足某些自然的非分类条件时，并开发了来自…更直接几何学观点的距离函数的Morse理论（已发表在杂志上）。重要的是要表明，切割基因座CP具有一个良好的结构（惠特尼分层），并且临界点Q的索引是根据与P，Q相连的最小测量学的数量来定义的，Q，Q和通常的限制为4的限制Q的常见索引，其中包含Q光滑函数的Q光滑函数。 hope to continue to study how generic is the conditions among all Riemannian metrics on a given manifold.With respect to distance functions, K. Kiyohara determined the explicit structure of the cut locus and the conjugate locus of any point in ellipsoids (and some Liouville manifolds) from a view point of integrable geodesic flow with J. Itoh, and A. Katsuda studied the inverse problem of the Neumann boundary value problem, namely how to reconstruct the inner Riemann metric from the distance function to the boundary of a Riemannian manifold with boundary.As for geometric inequalities (isosystolic inequality, isodiametric inequality) and the first eigenvalue estimate of the Laplacian of a compact Riemannian manifolds with non-negative Ricci curvature and its perturbation, we could not make在此期间，基约哈拉（Kiyohara）获得了有关亚历山德罗夫（Alexandrov）空间几何不平等的结果。希望继续进行该计划。 Y. Mori研究了量子计算机的算法，并支持Sakai在计算机辅助方面。较少的