Cohomologie feuillet\'ee du flot affine de Reeb sur la vari\'et\'e de Hopf ${\Bbb S}^n\times {\Bbb S}^1$

TL;DR

This paper explicitly computes the foliated cohomology of the affine Reeb flow on Hopf manifolds, revealing the obstructions to solving cohomological equations and characterizing invariant distributions.

Contribution

It provides an explicit description of the foliated cohomology for the affine Reeb flow on Hopf manifolds, linking cohomology classes to obstructions and invariant distributions.

Findings

Explicit computation of $H_{\

f}^1(M)$ for the affine Reeb flow

Identification of obstructions to solving cohomological equations as cohomology classes,

Abstract

We determine explicitly the foliated cohomology $H_{\cal F}^\ast (M)$ of the affine Reeb flow ${\cal F}$ on the Hopf manifold ${\Bbb S}^n\times {\Bbb S}^1$. The vector space $H_{\cal F}^1(M)$ contains exactly the obstructions to solve the cohomological equation $X\cdot f=g$ where $f$ and $g$ are $C^\infty $-functions and $X$ is any non singular vector field defining the foliation ${\cal F}$. The topological dual of $H_{\cal F}^1(M)$ is the space of distributions invariant by $X$.