Vertical Generalized Berger-type Metrics: A New Perspective on Harmonic Vector Fields

In this paper, we introduce a metric called the vertical generalized Berger-type deformed Sasaki metric, defined on the tangent bundle of an anti-paraK"{a}hler manifold. First, we analyze the harmonicity of vector fields with respect to this new metric, providing examples that illustrate how certain vector fields satisfy the harmonicity condition under the introduced metric. These examples demonstrate the unique properties and behavior of harmonic vector fields on anti-paraK"{a}hler manifolds equipped with this specific metric. Next, we explore the harmonicity of a vector field along a map between Riemannian manifolds, where the target manifold is anti-paraK"{a}hler and its tangent bundle is equipped with the vertical generalized Berger-type deformed Sasaki metric. Finally, we investigate the harmonicity of the composition of two specific maps. The first map is the projection from the tangent bundle of a Riemannian manifold (the source manifold) onto the manifold itself, and the second map is from this Riemannian manifold to another Riemannian manifold. The source manifold is an anti-paraK"{a}hler manifold, and its tangent bundle is endowed with the vertical generalized Berger-type deformed Sasaki metric. We discuss the conditions under which the composition of these maps produces harmonic vector fields.