Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Variations of the Godbillon--Vey invariant of transversely parallelizable foliations (1909.13250v1)

Published 29 Sep 2019 in math.DG

Abstract: We consider a $(2q+1)$-dimensional smooth manifold $M$ equipped with a $(q+1)$-dimensional, a priori non-integrable, distribution ${\cal D}$ and a $q$-vector field ${\bf T}=T_1\wedge\ldots\wedge T_q$, where ${T_i}$ are linearly independent vector fields transverse to~${\cal D}$. Using a $q$-form $\omega$ such that ${\cal D} = \ker\,\omega$ and $\omega({\bf T})=1$, we construct a $(2q+1)$-form analogous to that defining the Godbillon--Vey class of a $(q+1)$-dimensional foliation, and show how does this form depend on $\omega$ and ${\bf T}$. For a compatible Riemannian metric $g$ on $M$, we express this $(2q+1)$-form in terms of ${\bf T}$ and extrinsic geometry of~${\cal D}$ and normal distribution ${\cal D}\bot$. We find Euler-Lagrange equations of associated functionals: for variable $(\omega,{\bf T})$ on $(M,g)$, and for variable metric on $(M,{\cal D})$, when distributions/foliations and forms are defined outside a "singularity set" under additional assumption of convergence of certain integrals. We show that for a harmonic distribution ${\cal D}\bot$ such $(\omega,{\bf T})$ is critical, characterize critical pairs when ${\cal D}$ is integrable and find sufficient conditions for critical pairs when variations are among foliations, calculate the index form and consider examples of critical foliations among twisted products, Reeb foliations and transversely holomorphic flows.

Summary

We haven't generated a summary for this paper yet.