Actions with globally hypoelliptic leafwise Laplacian and rigidity

Abstract

We prove several results concerning smooth ${\mathbb{R}}^k$ actions with the property that their leafwise Laplacian is globally hypoelliptic. Such actions are necessarily uniquely ergodic and minimal, and cohomology is often finite-dimensional, even trivial. Further we consider a class of examples of ${\mathbb{R}}^2$ actions on 2-step nilmanifolds, which have globally hypoelliptic leafwise Laplacian, and we show transversal local rigidity under certain Diophantine conditions.