Some harmonic problems for deformed Sasaki metrics on tangent bundles

We investigate some harmonic problems on the tangent bundle with a deformed Sasaki metric. Firstly, we study the harmonicity of a vector field with respect to this metric, and we construct some examples of harmonic vector fields. Secondly, we study the harmonicity of a vector field along a map between Riemannian manifolds, where the tangent bundle of the target manifold is equipped with a deformed Sasaki metric. Finally, we discuss the harmonicity of the composition of the projection map of the tangent bundle of a Riemannian manifold with a map from this manifold into another Riemannian manifold, where the tangent bundle of the first manifold is equipped with a deformed Sasaki metric.