On global hypoellipticity of horizontal Laplacians on compact principal bundles (Q810978)

Let M be a smooth, Riemannian, connected manifold without boundary and let \(X_ 1,X_ 2,...,X_ k\) be finitely many smooth vector fields on M. The author states the following conjecture: A differential operator \(L=\sum X^*_ iX_ i\) is globally hypoelliptic if the group generated by \(\{\) exp \(\sum f_ iX_ i\); \(f_ i\in C^{\infty}(M)\}\) acts transitively on M. This conjecture is affirmative in the real analytic case because of the Hörmander theorem. In the paper the case of horizontal Laplacian on compact principal bundles is considered and a characterization of its global hypoellipticity is given.