Harmonic maps on the tangent bundle according to the ciconia metric

The focus of this paper revolves around investigating the harmonicity aspects of various mappings. Firstly, we explore the harmonicity of the canonical projection pi: (TM, (g) over tilde) -> (M-2n, J, g), where (M-2n, J, g) represents an anti-paraKahler manifold and (TM, (g) over tilde) its tangent bundle with the ciconia metric. Additionally, we study the harmonicity of a vector field, treated as mappings from M to TM. In this context, we consider the harmonicity relations between the ciconia metric (g) over tilde and the Sasaki metric Sg, examining their mutual interactions. Furthermore, we investigate the Schoutan-Van Kampen connection and the Vranceanu connection, both associated with the Levi-Civita connection of the ciconia metric. Our analysis also includes the computation of the mean connections for the Schoutan-Van Kampen and Vranceanu connections, thereby providing insights into their properties. Finally, our exploration extends to the second fundamental form of the identity ((mapping from (TM, (g) over bar) to TM, (del) over bar (m)) and TM, (del) over tilde *m). Here (del) over bar (m) and (del) over tilde (*m) denote the mean connections associated with the Schoutan-Van Kampen and Vranceanu connections, respectively.