Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
A Stochastic approach to the Poincaré-Hopf theorem
Download PDF
Download PDF
  • Published: June 1989

A Stochastic approach to the Poincaré-Hopf theorem

  • Naomasa Ueki1 

Probability Theory and Related Fields volume 82, pages 271–293 (1989)Cite this article

  • 115 Accesses

  • 4 Citations

  • Metrics details

Summary

In this paper, the Poincaré-Hopf theorem (Hopf [8]) is proved by a probabilistic method. The proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution. We use the Malliavian calculus to estimate the fundametal solution. Our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes: II. Ann. Math., II. Ser.88, 451–491 (1968)

    Google Scholar 

  2. Bismut, J.-M.: The Atiyah-Singer theorems: A probabilistic approach, I; the index theorem. J. Funct. Anal.57, 56–99 (1984)

    Google Scholar 

  3. Bismut, J.-M.: The Witten complex and the degenerate Morse inequalities. J. Differ. Geom.23, 207–240 (1986)

    Google Scholar 

  4. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger operators. Berlin Heidelberg New York: Springer 1987

    Google Scholar 

  5. Dold, A.: Fixed point index and fixed point theorem for Euclidean neighborhood retracts. Topology4, 1–8 (1965)

    Google Scholar 

  6. Gilkey, P.B.: Invariance Theory, the heat equation, and the Atiyah-Singer Index theorem. Wilmington Delaware: Publish or Perish, Inc. 1984

    Google Scholar 

  7. Guilemin, V., Pollack, A.: Differential topology. Englewood Cliffs New Jersey: Princeton-Hall 1974

    Google Scholar 

  8. Hopf, H.: Vectorfelder inn-dimensionalen Mannigfaltigkeiten. Mat. Ann.96, 225–250 (1927)

    Google Scholar 

  9. Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Amsterdam Tokyo: North Holland-Kodansha 1981

    Google Scholar 

  10. Ikeda, N., Watanabe, S.: Malliavin calculus of Wiener functionals and its applications. In: Elworthy, K.D. (ed.). From local times to global geometry, control and physics. (Pitman Research Notes in Math. Series, vol. 150, pp. 132–178). Harlow: Longman Scientific & Technical 1987

    Google Scholar 

  11. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. I. New York: Interscience 1963

    Google Scholar 

  12. Milnor, J.W.: Topology from the differential viewpoint. The University Press of Virginia 1965

  13. Spivak, M.: A comprehensive Introduction to Differential Geometry V. Boston: Publish or Perish, Inc. 1975

    Google Scholar 

  14. Watanabe, S.: Lectures on stochastic differential equations and Malliavin calculus. Berlin Heidelberg New York: Tata Institute of Fundamental Research, Springer 1984

    Google Scholar 

  15. Watanabe, S.: Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab.15, 1–39 (1987)

    Google Scholar 

  16. Watanabe, S.: Generalized Wiener functionals and their applications. In: Watanabe, S., Prochorov, Y.V. (eds.). Probability theory and mathematical statistics. Proc. Fifth Japan-USSR Sympos. on Probab., Kyoto 1986. (Lect. Notes., vol. 1299, pp. 541–548). Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  17. Witten, E.: Supersymmetry and Morse theory. J. Differ. Geom.17, 661–692 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Osaka University, 560, Toyonaka, Osaka, Japan

    Naomasa Ueki

Authors
  1. Naomasa Ueki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ueki, N. A Stochastic approach to the Poincaré-Hopf theorem. Probab. Th. Rel. Fields 82, 271–293 (1989). https://doi.org/10.1007/BF00354764

Download citation

  • Received: 01 July 1988

  • Issue Date: June 1989

  • DOI: https://doi.org/10.1007/BF00354764

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Fundamental Solution
  • Probabilistic Method
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature