In this talk, we first show the CR Lichnerowicz theorem for Kohn Laplacian in a closed
pseudohermitian (2n+1)- manifold (M,θ). Secondly, we confirm the corresponding CR Obata theorem
in a closed (2n+1)-manifold M with a volume-normalized contact form θ. In particular, it is true
for a closed pseudohermitian hypersurface in C^{n+1}. 
This is a join work with Prof. Shu-Cheng Chang.