| Co-Investigator(Kenkyū-buntansha) |
KATSUDA Atsushi OKAYAMA UNIVERSITY, FACULTY OF SCIENCE, ASSOCIATE PROFESSOR, 理学部, 助教授 (60183779)
TAMURA Hideo OKAYAMA UNIVERSITY, FACULTY OF SCIENCE, PROFESSOR, 理学部, 教授 (30022734)
KIYOHARA Kazuyoshi OKAYAMA UNIVERSITY, FACULTY OF SCIENCE, PROFESSOR, 理学部, 教授 (80153245)
SHIOYA Takashi TOHOKU UNIVERSITY, GRADUATE SCHOOL OF SCIENCE, ASSOCIATE PROFESSOR, 大学院・理学研究科, 助教授 (90235507)
KASUE Atsushi KANAZAWA UNIVERSITY, FACULTY OF SCIENCE, PROFESSOR, 理学部, 教授 (40152657)
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| Research Abstract |
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
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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
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