• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to project page

2003 Fiscal Year Final Research Report Summary

Relations between space-structures and curvatures

Research Project

Project/Area Number 12440020
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionOKAYAMA UNIVERSITY

Principal Investigator

SAKAI Takashi  OKAYAMA UNIVERSITY, FACULTY OF SCIENCE, PROFESSOR, 理学部, 教授 (70005809)

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)
Project Period (FY) 2000 – 2003
KeywordsRiemannian manifolds / Alexandrov spaces / Curvatures / Metric Invariants / Distance functions
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 … 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

  • Research Products

    (28 results)

All Other

All Publications (28 results)

  • [Publications] Takashi Sakai: "Warped products, Ricci curvature and distance functions"Tohoku Math.Publications(Proc.of the Fifth Pacific Rim Geometr Conference). 20. 163-172 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi Sakai: "Curvature -- until the twentieth century, and the future?"Sugaku Exposition (Amer.Math.Soc.). (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kazuyoshi Kiyohara: "Two-dimensional geodesic flows having first integrals of higher degree"Math.Ann.. 320. 487-505 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Kazuyoshi Kiyohara: "On Kaehler-Liouville manifolds"Contemp.Math.. 308. 211-222 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] T.Ichinose, Hideo Tamura: "The norm convergence of the Trotter-Kato product formula with error bound"Comm.Math.Phys.(2001). 217. 489-502 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Hideo Tamura: "Resolvent convergence in norm for Dirac operator with Aharonov-Bohm field"J.Math.Phys.. 44. 2967-2993 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Katsuda, Y.V.Kurylev, M.Lassas: "Stability and Reconstction in Gel'fand Inverse Boundary Spectral Problem"New Analytic and Geometric Methods In Inverse Problems (K.Bingham, Ya.V.Kurylev, E.Sommesalo eds.). 309-322 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] M.T.Anderson, A.Katsuda, Y.Kurylev, M.Lassas, M.E.Taylor: "Metric tensor estimate, geometric convergence, and inverse boundary problem"Electron.Res.Announc.Amer.Math.Soc.. 9. 69-79 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Kasue: "Convergence of Riemannian manifolds and Laplace operators I"Ann.I'Institut Fourier. 52. 1219-1257 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Atsushi Kasue, H.Kumura: "Spectral convergence of conformally immersed surfaces with bounded mean curvature"J.Geom.Anal.. 12. 663-681 (2002)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Kuwae, Takashi Shioya: "Sobolev and Dirichlet spaces over maps between metric spaces"J.Reine Angew.Math.. 555. 39-75 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi Shioya, T.Yamaguchi: "Volume collapsed three-manifolds under a lower curvature bound"Math.Ann. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Shiohama, Takashi Shioya, M.Tanaka: "The geometry of total curvature on complete open surfaces"Cambridge Univ.Press, Cambridge. 284 (2003)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] 加須栄 篤: "リーマン幾何学"培風館. 264 (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Takashi Sakai: "Warped products, Ricci curvature and distance functions"Tohoku Math.Publications (Proc.of the Fifth Pacific Rim Geometry Conference). 20. 163-172 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Sakai: "Curvature --until the twentieth century, and the future?"Sugaku Exposition. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuyoshi Kiyohara: "Two dimensional geodesic flows having first integrals of higher degree"Math.Ann.. 320. 487-505 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Kazuyoshi Kiyohara: "On Kaehler-Liouville manifolds"Contemp.Math.. 308. 211-222 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] T.Ichinose, Hideo Tamura: "The norm convergence of the Trotter-Kato product formula with error bound"Comm.Math.Phys.. 217. 489-502 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Hideo Tamura: "Resolvent convergence in norm for Dirac operator with Aharonov-Bohm field"J.Math.Phys.. 44. 2967-2993 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atsushi Katsuda, Y.V.Kurylev, M.Lassas: "Stability and Reconstruction in Gel'fand Inverse Boundary Spectral Problem"New Analytic and Geometric Methods in Inverse Problems (K.Bingham, Ya.V.Kurylev, E.Sommesalo eds.). 309-322 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] M.T.Anderson, A.Katsuda, Y.Kurylev, M.Lassas M.E.Taylor: "Metric tensor estimate, geometric convergence, and inverse boundary problem"Electron.Res.Announc.Amer.Math.Soc.. 9. 69-79 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atsushi Kasue: "Convergence of Riemannian manifolds and Laplace operators I"Ann.l'Institut Fourier. 52. 1219-1257 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atsushi Kasue, H.Kumura: "Spectral convergence of conformally immersed surfaces with bounded mean curvature"J.Geom.Anal.. 12. 663-681 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Kuwae, Takashi Shioya: "Sobolev and Dirichlet spaces over maps between metric spaces"J.Reine Angew.Math.. 555(to appear in Math.Ann.). 39-75 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Takashi Shioya, T.Yamaguchi: "Volume collapsed three-manifolds under a lower curvature bound"Math.Ann.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K.Shiohama, Takashi Shioya, M.Tanaka: "The geometry of total curvature on complete open surfaces"Cambridge Univ. Press, Cambridge. 284 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Atushi Kasue: "Riemannian Geometry (in Japanese)"Baifukan Publ. C.. 264 (2001)

    • Description
      「研究成果報告書概要(欧文)」より

URL: 

Published: 2005-04-19  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi