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

This paper aims to study the Berger type deformed Sasaki metric g_BS 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 g_BS 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 \(T_{1,1}^{2}M\) $$T_{1,1}^{2}M$$ , 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, g_BS). Moreover, they search the harmonicity of the Berger type deformed Sasaki metric g_BS and the Sasaki metric g_S with respect to each other.