The second-order tangent bundle with deformed 2nd lift metric

Let (M,g) be a pseudo-Riemannian manifold and (TM)-M-2 be its second-order tangent bundle equipped with the deformed 2nd lift metric (g) over bar which is obtained from the 2nd lift metric by deforming the horizontal part with a symmetric (0, 2)-tensor field c. In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of ((TM)-M-2,(g) over bar). We give necessary and sufficient conditions for ((TM)-M-2,(g) over bar) to be semi-symmetric. Secondly, we show that ((TM)-M-2,(g) over bar) is a plural-holomorphic B-manifold with the natural integrable nilpotent structure. Finally, we get the conditions under which ((TM)-M-2,(g) over bar) with the 2nd lift of an almost complex structure is an anti-Kahler manifold.