Metric invariants and space structures

紧黎曼流形到点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d_p不可微的点，并且这个点包含在p的切轨迹C_p中。已知临界点的概念可以像通常的Morse理论一样引入，并且d_p在临界点处不可微。在没有临界点的情况下，莫尔斯理论（同位素引理）在光滑情况下得到了发展，在应用中发挥了重要作用。现在，为了发展包含临界点的距离函数的莫尔斯理论，我们应该首先定义临界点上的指数的概念。在本基金的支持下，本研究项目负责人T. Sakai与J. Itoh在黎曼度规满足一些自然非简并条件时进行了这一研究，并从更直接的几何角度发展了距离函数的Morse理论（已发表在期刊上）。那里显示是很重要的轨迹Cp携带一个好的结构(惠特尼分层),和一个临界点的指数q是定义的最小数量的测地线加入p, q和通常的指数q的限制4层包含q q是一个光滑函数的临界点。我希望能继续学习一般是如何在所有条件给定歧管黎曼度量。对于距离函数，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）中任意点的切割轨迹和共轭轨迹的显式结构，a . Katsuda研究了Neumann边值问题的反问题，即如何从距离函数到有边界的黎曼流形的边界重构内黎曼度规。对于几何不等式（等收缩不等式、等径不等式）和非负Ricci曲率紧致黎曼流形的拉普拉斯算子的第一特征值估计及其摄动，我们在这一时期没有取得实质性的进展，但是Kiyohara得到了关于Alexandrov空间几何不变量不等式的一个结果。我希望继续执行这个项目。Y. Mori研究了量子计算机的算法，并在计算机辅助中支持Sakai。少

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