## Electronic Communications in Probability

### A heat flow approach to the Godbillon-Vey class

Diego S. Ledesma

#### Abstract

We give a heat flow derivation for the Godbillon Vey class. In particular we prove that if $(M,g)$ is a compact Riemannian manifold with a codimension 1 foliation $\mathcal{F}$, defined by an integrable 1-form $\omega$ such that $||\omega ||=1$, then the Godbillon-Vey class can be written as $[-\mathcal{A} \omega \wedge d\omega ]_{dR}$ for an operator $\mathcal{A} :\Omega ^*(M)\rightarrow \Omega ^*(M)$ induced by the heat flow.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 2, 6 pp.

Dates
Accepted: 28 June 2015
First available in Project Euclid: 5 January 2017

https://projecteuclid.org/euclid.ecp/1483585771

Digital Object Identifier
doi:10.1214/16-ECP3836

Mathematical Reviews number (MathSciNet)
MR3607797

Zentralblatt MATH identifier
1358.58016

#### Citation

Ledesma, Diego S. A heat flow approach to the Godbillon-Vey class. Electron. Commun. Probab. 22 (2017), paper no. 2, 6 pp. doi:10.1214/16-ECP3836. https://projecteuclid.org/euclid.ecp/1483585771

#### References

• [1] Elworthy, K. D. Stochastic Differential Equations on Manifolds. London Mathematical Society Lecture Note Series, 70. Cambridge University Press, Cambridge-New York, 1982.
• [2] Elworthy, K. D.; Rosenberg, S. Homotopy and homology vanishing theorems and the stability of stochastic flows. Geom. Funct. Anal. 6 (1996), no. 1, 51–78.
• [3] Elworthy, K. D.; Le Jan, Y.; Li, Xue-Mei. On the geometry of diffusion operators and stochastic flows. Lecture Notes in Mathematics, 1720. Springer-Verlag, Berlin, 1999.
• [4] Godbillon, C. and Vey, J. Un invariant des feuilletages de codimension 1, C. R. Acad. Sci. Paris Sér. A-B 273 (1971) A92–A95.
• [5] Hurder, S. Dynamics and the Godbillon-Vey class: a history and survey. Foliations: geometry and dynamics (Warsaw, 2000), 29–60, World Sci. Publ., River Edge, NJ, 2002.
• [6] N. Ikeda and S. Watanabe Stochastic differential equations and diffusion processes. Second edition. North-Holland Mathematical Library, 24. North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.
• [7] Kunita, H. Some extensions of Itô’s formula. Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), pp. 118–141, Lecture Notes in Math., 850, Springer, Berlin, 1981.
• [8] D. Sullivan. Cycles for the dynamical study of foliated manifolds and complex manifolds. Invent. Math. 36, 225–265 (1976).