Nonnegativity of CR Paneitz Operator and Its Application to the CR Obata's Theorem

In this paper, we first prove the CR analogue of M. Obata's theorem on a closed pseudohermitian (2n+1)-manifold with free pseudohermitian torsion. Secondly, we have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian on a closed pseudohermitian (2n+1)-manifold with a more general curvature condition for na parts per thousand yen2. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator P (0) for na parts per thousand yen2.