The diffeomorphism group of a Lie foliation

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.