Notes on the tangent bundles over $F$-Kählerian manifolds

Let $TM$ be the tangent bundle over an $F$-Kählerian manifold endowed with Berger type deformed Sasaki metric $g_{BS}$. In this paper, we obtain the Levi-Civita connection of this metric and study geodesics on the tangent bundle $TM$ and $F$-unit tangent bundle $T_{1,F}M.$ Secondly, we characterize the geodesic curvatures on $T_{1,F}M$. Finally, we present some conditions for a vector field $\xi :M\rightarrow TM$ to be harmonic and study the harmonicity of the canonical projection $\pi :TM\rightarrow M$. In addition, we search the harmonicity of the Berger type deformed Sasaki metric $g_{BS}$ and the Sasaki metric $g_{S}$ with respect to each other.