The diffeomorphism group of a Lie foliation

2008/12/13 by G. Hector, Hector, G., E. Macías-Virgós +3
Mathematics · #22E65 #57R30 #58B25 #58D05 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:22E65 #msc:57R30 #msc:58B25 #msc:58D05

paper · pdf · doi:10.48550/arxiv.0812.2550

16 pages

arxiv created 2008/12/13 · arxiv updated 2009/12/01

Abstract

We explicitly compute the diffeomorphism group of several types of linear foliations (with dense leaves) on the torus Tⁿ, n≥ 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 ± \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 T², P. Iglesias and G. Lachaud for codimension one foliations on Tⁿ, n≥ 2, and B. Herrera for transcendent foliations. The theoretical setting of the paper is that of J. M. Souriau's diffeological spaces.

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