Bochner-Type Formulas for Transversally Harmonic Maps

Abstract

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 T f is at least three at some points of M , then f is contact holomorphic (or contact anti-holomorphic).

Harmonic maps on weighted foliations

“…Let Q be the normal bundle of F and d T φ = dφ| Q , the restriction of dφ on the normal bundle Q. Then φ is said to be transversally harmonic if φ is a solution of the Eular-Largrange equation τ b (φ) = 0, where τ b (φ) = tr Q (∇ tr d T φ) is the transversal tension field of φ. Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [4,10,13,14,15,19,20,30]. However, a transversally harmonic map is not a critical point of the transversal energy functional [14]…”

Liouville type theorem for (F;F')p-harmonic maps on foliations

“…Then φ is said to be transversally harmonic if the transversal tension field τ b (φ) = tr Q (∇ tr d T φ) vanishes, where d T φ = dφ| Q and Q is the normal bundle of F . Transversally harmonic maps on foliated Riemannian manifolds have been studied by many authors [3,12,13,18]. However, a transversally harmonic map is not a critical point of the transversal energy [10]…”