Infinitesimally rigid Lie foliations with dense leaves

Abstract

We call a foliation $\mathcal{F}$ on a compact manifold infinitesimally rigid if its deformation cohomology $H^{1}(\mathcal{F},N\mathcal{F})$ vanishes. This paper studies infinitesimal rigidity for a distinguished class of Riemannian foliations, namely Lie foliations with dense leaves. We construct infinitesimally rigid Lie foliations with dense leaves, modeled on any compact semisimple Lie algebra with simple ideals different from $\mathfrak{so}(3)$. To our knowledge, these are the first examples of infinitesimally rigid Riemannian foliations that are not Hausdorff.