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

Let [Formula: see text] be a pseudo-Riemannian manifold and [Formula: see text] be its second-order tangent bundle equipped with the deformed [Formula: see text]nd lift metric [Formula: see text] which is obtained from the [Formula: see text]nd lift metric by deforming the horizontal part with a symmetric [Formula: see text]-tensor field [Formula: see text]. In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of [Formula: see text]. We give necessary and sufficient conditions for [Formula: see text] to be semi-symmetric. Secondly, we show that [Formula: see text] is a plural-holomorphic [Formula: see text]-manifold with the natural integrable nilpotent structure. Finally, we get the conditions under which [Formula: see text] with the [Formula: see text]nd lift of an almost complex structure is an anti-Kähler manifold.