Schwarz Type Lemmas for Pseudo-Hermitian Manifolds

In this paper, we consider some generalized holomorphic maps between pseudo-Hermitian manifolds. These maps include the CR maps and the transversally holomorphic maps. In terms of some sub-Laplacian or Hessian type Bochner formulas, and comparison theorems in the pseudo-Hermitian version, we are able to establish several Schwarz type results for both the CR maps and the transversally holomorphic maps between pseudo-Hermitian manifolds. Finally, we also discuss the CR hyperbolicity problem for pseudo-Hermitian manifolds.