almost complex structure over almost contact metric structures

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 k¨ahlerian properties. we establish several characterizations relating the geometry of (tm, j) to the cosymplectic structure on m. specifically, we show that (tm, j) is k¨ahlerian if and only if (m, η, ξ, φ) is cosymplectic and r = 0. similarly, we prove that (tm, j) is nearly k¨ahlerian under the same conditions on m. furthermore, we present an alternative criterion for (tm, j) to be k¨ahlerian, involving a nearly cosymplectic condition on m alongside a specific curvature relation. finally, we demonstrate that (tm, j) is semi-k¨ahlerian if and only if (m, η, ξ, φ) is semi-cosymplectic with r(x, y )φz = 0. these results reveal intricate connections between cosymplectic structures on m and k¨ahlerian-type structures on tm, contributing to the broader understanding of almost complex geometry on tangent bundles.