The Bochner-type formula and the first eigenvalue of the sub-Laplacian on a contact Riemannian manifold

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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 Riemannian case.