The Lichnerowicz-Obata theorem on sub-Riemannian manifolds with   transverse symmetries
  We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian sphere. 
Fabrice Baudoin, Bumsik Kim
math.DG