Bochner-type formulas for transversally harmonic maps

The main purpose of this paper is to study the properties of transversally harmonic maps by using Bochner-type formulas. As an application, we obtain the following theorem between compact Sasaki manifolds: Let f be a transversally harmonic map from compact Sasaki manifold M to compact Sasaki manifold M′, and M′ has a strongly negative transverse curvature. If the rank of d _Tf is at least three at some points of M, then f is contact holomorphic (or contact anti-holomorphic).