  Contact Riemannian manifolds, with not necessarily integrable complex
structures, are the generalization of pseudohermitian manifolds in CR geometry.
The Tanaka-Webster-Tanno connection on such a manifold plays the role of
Tanaka-Webster connection in the pseudohermitian case. We prove the contact
Riemannian version of the pseudohermitian Bochner-type formula, and generalize
the CR Lichnerowicz theorem about the sharp lower bound for the first nonzero
eigenvalue of the sub-Laplacian to the contact Riemannnian case.
