A Lichnerowicz estimate for the spectral gap of the sub-Laplacian

For a second order operator on a compact manifold satisfying the strong H"ormander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.