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 π: (T M, ˜g) → (M2n, J, g), where (M2n, J, g) represents an anti-paraKähler manifold and (T M, ˜g) its tangent bundle with the ciconia metric. Additionally, we study the harmonicity of a vector field ξ, treated as mappings from M to T M. In this context, we consider the harmonicity relations between the ciconia metric ˜g and the Sasaki metricSg, examining their mutual interactions. Furthermore, we investigate the Schoutan-Van Kampen connection and the Vrãnceanu 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 Vrãnceanu connections, thereby providing insights into their properties. Finally, our exploration( extends to the second fundamental form of the identity mapping from (T M, ˜g) to T M, ∇m) ( and T M,˜∇∗m). Here ∇m and˜∇∗m denote the mean connections associated with the Schoutan-Van Kampen and Vrãnceanu connections, respectively.