Available online xxxx Communicated by F. Thomas MSC: 53D10 32V20 53C17Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The TanakaWebster-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 subLaplacian to the contact Riemannian case.
Contact Riemannian manifolds, whose complex structures are not necessarily integrable, are generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection plays the role of the Tanaka-Webster connection of a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By constructing the special frames and the normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, its 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.
Contents1. Introduction 2. Construction of the normal coordinates 2.1. The TWT connection 2.2. The structure equations 2.3. The special frame and the normal coordinates 2.4. Homogeneous parts of the special frame (coframe) and the connection coefficients 2.5. The asymptotic expansion of the special frame 3. The asymptotic expansion of the almost complex structure, the curvature and Tanno tensors 3.1. The Tanno tensor at point q 3.2. The asymptotic expansion of the almost complex structure at point q 3.3. The asymptotic expansion of curvature tensors 4. The normalized special frame 4.1. The transformation formulae under the conformal transformation 4.2. The conformal contact form with vanishing R γ α γ β (q) and A αβ (q) 5. The proof of the main theorem 5.1. The asymptotic expansion of the Yamabe functional 5.2. Calculation of some integrals 5.3. Calculation of constants a m (n) and b m (n) Appendix A. The transformation formulae
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