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2010 Fiscal Year Final Research Report

Study on the general adiabatic expansion theory for heat kernels and its some applications

Research Project

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Project/Area Number 20540063
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Geometry
Research InstitutionSaitama University

Principal Investigator

NAGASE Masayoshi  Saitama University, 大学院・理工学研究科, 教授 (30175509)

Co-Investigator(Renkei-kenkyūsha) MIZUTANI Tadayoshi  埼玉大学, 名誉教授 (20080492)
SAKAMOTO Kunio  埼玉大学, 大学院・理工学研究科, 教授 (70089829)
SHIMOKAWA Koya  埼玉大学, 大学院・理工学研究科, 准教授 (60312633)
FUKUI Toshisumi  埼玉大学, 大学院・理工学研究科, 教授 (90218892)
Project Period (FY) 2008 – 2010
Keywords熱核 / 断熱展開 / 漸近展開
Research Abstract

We clarified the usefulness and powerfulness of the general adiabatic expansion theory in studying some subjects related to the heat kernels. In particular, based on the theory we obtained a formula for the coefficients of the asymptotic expansion of every derivative of the heat kernel associated to the Riemannian Laplacian. In order to describe the coefficients explicitly up to an arbitrarily high order as universal polynomials built from the curvature, we need only a basic knowledge of calculus added to the formula. (The conventional method requires various kinds of profound knowledge so that only a few coefficients were investigated.) We found also a relation between the adiabatic expansion and certain anomalies which are often referred to by physicists, and introduced an interesting formula for the anomalies.

  • Research Products

    (3 results)

All 2010 2009

All Journal Article (3 results) (of which Peer Reviewed: 3 results)

  • [Journal Article] Expressions of the heat kernels on spheres by elementary functions and their recurrence relations2010

    • Author(s)
      長瀬正義
    • Journal Title

      Saitama Math.J. Vol.27

      Pages: 23-34

    • Peer Reviewed
  • [Journal Article] The Laplacian and the heat kernel acting on differential forms on spheres2009

    • Author(s)
      長瀬正義
    • Journal Title

      Tohoku Math.J. Vol.61

      Pages: 571-588

    • Peer Reviewed
  • [Journal Article] Finite surgeries on three-tangle pretzel knots2009

    • Author(s)
      下川航也,他
    • Journal Title

      Algebr.Geom.Topol. Vol.9

      Pages: 743-771

    • Peer Reviewed

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Published: 2012-01-26   Modified: 2016-04-21  

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