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

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

紧致黎曼流形到点p的距离函数d_p是一个重要的几何函数，它也与流形结构有关。然而，距离函数d_p允许一个d_p不可微的点，而这个点包含在p的割轨迹C_p中。我们知道，可以像通常的Morse理论那样引入临界点的概念，d_p在临界点是不可微的。在不存在临界点的情况下，Morse理论(等价点引理)已经发展为光滑情形，在应用中发挥了重要作用。现在，要发展包含临界点的距离函数的Morse理论，首先应该定义临界点上的指数的概念。在本研究计划的首席研究员T.Sakai的支持下，当Riemannian度量满足一些自然的非退化条件时，T.Sakai与J.Itoh一起完成了这项工作，并从…发展了距离函数的莫尔斯理论更直接的几何观点(已在期刊上发表)。在这里，重要的是证明割轨迹Cp具有良好的结构(惠特尼分层)，并且临界点Q的指数是根据连接p，q的极小测地线的数目和Q处的通常指数来定义的，该指数限制在包含Q的层上，该层是具有临界点q的光滑函数。我希望继续研究给定流形上的所有黎曼度量之间的条件是如何通用的。关于距离函数，K.Kiyohara和J.Itoh从可积测地流的角度确定了椭球体(和一些Liouville流形)中任何一点的截线和共轭轨迹的显式结构，A.Katsuda研究了Neumann边值问题的反问题，即如何重构从距离函数到有边界的黎曼流形的内黎曼度量。关于几何不等式(等晶不等式，等径不等式)和具有非负Ricci曲率及其扰动的紧致黎曼流形的Laplace算子的第一特征值估计，在这一时期，我们不能取得实质性的进展但是Kiyohara得到了一个关于Alexandrov空间的几何不变量的不等式的结果。我希望继续进行这个项目。Y.Mori研究了量子计算机的算法，并在计算机辅助方面支持Sakai。较少

项目成果

Curvature-up through the twentieth century,and its future

二十世纪的曲率上升及其未来

• 发表时间: 2005
• 期刊: Sugaku Expositions 18
• 作者: J.Itoh;C.Vilcu;酒井 隆;T.Sakai
• 通讯作者: T.Sakai

Geodesic characterization of the isosceles tetrahedron

等腰四面体的测地线表征

• 发表时间: 2007
• 作者: J. Itoh;T. Zamfirescu;伊藤 仁一;J. Itoh;清原 一吉;K. Kiyohara;清原 一吉;勝田 篤;A. Katsuda;勝田 篤;A. Katsuda;伊藤 仁一
• 通讯作者: 伊藤 仁一

Cut loci and Jacobi fields of Liouville manifolds

刘维尔流形的割轨迹和雅可比场

• 发表时间: 2006
• 作者: J. Itoh;C. Vilcu;伊藤 仁一;J. Itoh;酒井 隆;T. Sakai;伊藤 仁一;J. Itoh;清原 一吉;K. Kiyohara;伊藤 仁一;J. Itoh;清原 一吉;K. Kiyohara;清原 一吉;K. Kiyohara
• 通讯作者: K. Kiyohara

Several topics of space curves and geodesics

空间曲线和测地线的几个专题

• 发表时间: 2006
• 作者: K. Enomoto;J. Itoh;R. Sinclair;伊藤 仁一
• 通讯作者: 伊藤 仁一

距離関数のMorse理論とcut locus

距离函数和切割轨迹的莫尔斯理论

• 发表时间: 2005
• 作者: J. Itoh;C. Vilcu;清原 一吉;K. Kivohara.;清原 一吉;K. Kiyohara;伊藤 仁一;J. Itoh;酒井 隆
• 通讯作者: 酒井 隆