On almost complex structures in the cotangent bundle

E. M. Patterson and K. Yano studied vertical and complete lifts of tensor fields and connections from a manifold M_n to its cotangent bundle T^{ast} (M_n). Afterwards, K. Yano studied the behavior on the cross-section of the lifts of tensor fields and connections on a manifold M_n to T^{ast} (M_n) and proved that when varphi defines an integrable almost complex structure on M_n, its complete lift ^C varphi is a complex structure. The main result of the present paper is the following theorem: Let varphi be an almost complex structure on a Riemannian manifold M_n. Then the complete lift ^C varphi of varphi, when restricted to the cross-section determined by an almost analytic 1-form omega on M_n, is an almost complex structure.