In this paper, we investigate the conditions under which a lifted almost complex structure J on the tangent bundle TM of a manifold M exhibits various Kählerian properties. We establish several characterizations relating the geometry of (TM, J) to the cosymplectic structure on M. Specifically, we show that (TM, J) is Kählerian if and only if (M, , , ) is cosymplectic and R = 0. Similarly, we prove that (TM, J) is nearly Kählerian under the same conditions on M. Furthermore, we present an alternative criterion for (TM, J) to be Kählerian, involving a nearly cosymplectic condition on M alongside a specific curvature relation. Finally, we demonstrate that (TM, J) is semi-Kählerian if and only if (M, , , ) is semi-cosymplectic with R (X, Y) Z = 0. These results reveal intricate connections between cosymplectic structures on M and Kählerian-type structures on TM, contributing to the broader understanding of almost complex geometry on tangent bundles.
Sadighi et al. (Wed,) studied this question.