Remarks about the Kaluza-Klein metric on tangent bundle

Abstract

The paper is concerned with the Kaluza–Klein metric on the tangent bundle over a Riemannian manifold. All kinds of Riemann curvature tensors are computed and some curvature properties are given. The compatible almost complex structure is defined on the tangent bundle, and necessary and sufficient conditions for such a structure to be integrable are described. Then, the condition is given under which the tangent bundle with these structures is almost Kähler. Finally, almost golden complex structures are defined on this setting and some results related to them are presented.