Almost complex structure over almost contact metric structures

Document Type : Original Article

Authors

1 Department of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran

2 Department of Mathematics, Bijar Branch, Islamic Azad University, Bijar, Iran

Abstract

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,η,ξ,φ)‎ 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,η,ξ,φ)‎ is semi-cosymplectic with ‎R(X,Y)φZ=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.

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