Notes on the tangent bundle with deformed complete lift metric

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1. X, ˜Y, ˜Z∈ ℑ1(T M ). We then obtain the following equations: 0for all ˜Y, ˜Z∈ ℑ1 (ΦHJGf)(VX,VY,HZ) 0, (ΦHJGf)(VX,VY,VZ) 0, (ΦHJGf)(VX,HY,VZ) 0, (ΦHJG)(VX,HY,HZ) g((∇YJ )X, Z) + g(Y, (∇ZJ )X), (ΦHJGf)(HX,VY,HZ) (ΦJg)(X, Y, Z)− g((∇YJ )X, Z), (ΦHJGf)(HX,VY,VZ) 0, (ΦHJGf)(HX,HY,HZ) = (J X)(f )g(Y, Z)− X(f)g(JY, Z) + f((ΦJg)(X, Y, Z)) +g(J R(Y, X)u− R(Y, JX)u, Z) +g(Y, J R(Z, X)u− R(Z, JX)u), (ΦHJG)(HX,HY,VZ) = (ΦJg)(X, Y, Z)− g(Y, (∇ZJ )X). It is well known that the equation ΦJg = 0 is equivalent to∇J = 0, and the Riemann curvature R of a K¨ahler-Norden manifold is totally pure. Therefore, from the equations above, we have the following result. Theorem 5.2 Let (M, g) be a pseudo-Riemannian manifold and T M be its tangent bundle equipped with the deformed complete lift metric Gfand the almost complex structureHJ. The triple ( T M,HJ, Gf is a K¨ahler Norden manifold if and only if the triple (M, J, g) is a K¨ahler-Norden manifold and the function f satisfies the condition (J X)(f )g(Y, Z)− X(f)g(JY, Z) = 0.