Some Research Of The Geometry And Analysis On Contact Riemannian Manifolds

Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno con-nection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. The sub-Laplacians, conformal transformations and the Yamabe problem are also defined naturally in this setting. Here we investigate the Lichnerowicz theorem and the Yamabe problem on contact Riemannian manifolds.In Chapter 1, we introduce the history and recent development of CR manifolds and contact Riemannian manifolds. And then we introduce the main results and methods of Lichnerowicz-type theorem and the Yamabe problem on contact Riemannian manifolds.In Chapter 2, We prove the contact Riemannian version of the pseudohermitian Bochner-type formula, and generalize the CR Lichnerowicz-type theorem about the sharp lower bound for the first nonzero eigenvalue of the sub-Laplacian to the contact Riemannnian case.In Chapter 3, we investigate the Yamabe problem on contact Riemannian manifolds. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, the Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. So the Yamabe problem on a contact Riemannian manifold is always solvable.