Advertisement

Cohomology Groups of Harmonic Spinors on Conformally Flat Manifolds

  • Tosiaki Kori
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We shall investigate various properties of the sheaf of harmonic spinors N on C2 and, more generally, on conformally flat spin 4-manifolds. We prove the Runge approximation theorem on C2, and the vanishing of cohomologies; H 1 (C 2,N) = 0 and H 1(S 4, N) = 0. We shall introduce a concept of divisors of meromorphic spinors on a compact conformally flat spin 4-manifold, and give an analogy of Riemann-Roch theorem.

Keywords

Harmonic spinors conformally flat manifold Runge theorem Riemann-Roch theorem 

Mathematics Subject Classification (2000)

Primary 58G30 Secondary 53C21 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. Booss-Bavnbeck and K.P. Wojciechowski, Elliptic boundary value problems for Dirac operators. Springer Basel AG, Basel-Boston, 1993.CrossRefGoogle Scholar
  2. [2]
    F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. Pitman advanced publishing program, (1982).zbMATHGoogle Scholar
  3. [3]
    R. Delanghe and F. Brackx, Runge’s theorem in hypercomplex function theory, J. Approx. Theory 29 (1980), 200–211.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    O. ForsterLectures on Riemann surfaces, Springer-Verlag, 1981.CrossRefzbMATHGoogle Scholar
  5. [5]
    R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., 1965.zbMATHGoogle Scholar
  6. [6]
    N. Hitchin, Harmonic Spinors, Adv. in Math. 14 (1974).Google Scholar
  7. [7]
    L. Hörmander, An Introduction to Complex Analysis in Several Variables, D. Van Norstrand, 1967.Google Scholar
  8. [8]
    T. Kori, Spinor analysis on C 2 and on conformally flat 4-manifolds, Japanese J. of Math. 28–1 (2002), 1–30.MathSciNetGoogle Scholar
  9. [9]
    T. Kori, Index of the Dirac operator on S 4 and the infinite dimensional Grassmannian on S 3, Japanese J. of Math. 22–1 (1996), 1–36.MathSciNetGoogle Scholar
  10. [10]
    H.B. Lawson and M.L. Michelsohn, Spin Geometry, Princeton Univ. Press, 1989.zbMATHGoogle Scholar
  11. [11]
    J. RyanRunge approximation theorems in complex Clifford analysis together with some of their applications, J. of Functional Anal. 70 No.2 (1987), 221–253.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Tosiaki Kori
    • 1
  1. 1.Department of Mathematics School of Science and EngineeringWaseda UniversityTokyoJapan

Personalised recommendations