Abstract
The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection
π : TM → M, where M denotes a Riemannian manifold and TM its associated tangent bundle. Additionally, we delve into the harmonicity of vector fields ξ ∈ χ (M) , treated as mappings from M to TM. Moreover, our exploration extends to situations such as the case involving the map π : (TM, ˜g) → (M2n, J, g), where (M2n, J, g) represents an anti-paraKähler manifold and (TM, ˜g) its tangent bundle with the ciconia metric. In this context, we delve into the harmonicity relations between the ciconia metric ˜g and the Sasaki metric Sg, examining their mutual interactions. Furthermore, we delve into the Schoutan-Van Kampen connection and the Vranceanu connection, both associated with the Levi-Civita connection of the ciconia metric. We also undertake the computation of the mean connections for the Schoutan-Van Kampen and Vranceanu connections, thereby shedding light on their properties. Finally, our investigation extends to the second fundamental form of the identity mapping from
(TM, ˜g) to (TM,∇m) and (TM, ∇∗m). Here ∇m and ∇∗m represent the mean connections associated with the Schoutan-Van Kampen and Vranceanu connections, respectively.