Transversally Harmonic Maps And Related Bochner-type Formulas

In this paper, we mainly study the properties of transversally harmonic maps by using Bochner-tpye formulas. We establish the maximum principle and the unique continuation of transversally harmonic maps, and study the relationship between transversal harmonicity and ordinary harmonicity. We also derive the second variation formula for transversally harmonic maps.Making use of Bochner-type formulas, we get the transversal constant of transversally harmonic maps under some curvature assumptions of domain mani-folds and target manifolds. When the Riemannian foliation is Kahler, we consider the transversally holomorphicity of transversally harmonic maps. Especially, we get the following fact:Let f be a transversally harmonic map between compact Sasaki manifolds M and M', and M' has a strongly negative transverse curva-ture. If the the rank of dT f is at least 3 at some points of M, then f is contact holomorphic (or contact anti-holomorphic).