The diffeomorphism group of a Lie foliation

TL;DR

This paper explicitly computes the diffeomorphism groups of various linear Lie foliations on tori, revealing rigidity properties and general formulas applicable to dense leaf foliations on compact manifolds.

Contribution

It provides a general formula for the diffeomorphism group of Lie foliations with dense leaves, extending previous results to higher-dimensional tori and non-quadratic cases.

Findings

Non-quadratic foliations are rigid, allowing only trivial transverse diffeomorphisms.

Explicit computation of diffeomorphism groups for specific linear foliations on tori.

Generalization of previous results to higher dimensions and broader classes of foliations.

Abstract

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus $T^n$, $n\geq 2$, namely codimension one foliations, flows, and the so-called non-quadratic foliations. We show in particular that non-quadratic foliations are rigid, in the sense that they do not admit transverse diffeomorphisms other than $\pm \id$ and translations. The computation is an application of a general formula that we prove for the diffeomorphism group of any Lie foliation with dense leaves on a compact manifold. Our results generalize those of P. Donato and P. Iglesias for $T^2$, P. Iglesias and G. Lachaud for codimension one foliations on $T^n$, $n\geq 2$, and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau's diffeological spaces.