A Study on the Second Order Tangent Bundles over Bi-Kählerian Manifolds

This paper aims to study the Berger type deformed Sasaki metric gBS on the second order tangent bundle T2M over a bi-Kählerian manifold M. The authors firstly find the Levi-Civita connection of the Berger type deformed Sasaki metric gBS and calculate all forms of Riemannian curvature tensors of this metric. Also, they study geodesics on the second order tangent bundle T2M and bi-unit second order tangent bundle T1,12M, and characterize a geodesic of the bi-unit second order tangent bundle in terms of geodesic curvatures of its projection to the base. Finally, they present some conditions for a section σ: M → T2M to be harmonic and study the harmonicity of the different canonical projections and inclusions of (T2M, gBS). Moreover, they search the harmonicity of the Berger type deformed Sasaki metric gBS and the Sasaki metric gS with respect to each other.