Mustafa ÖZKAN, Aydın GEZER

Notes on the tangent bundle with deformed complete lift metric

ISSN: 1300-0098
Yayın Aralığı: 6
Yayıncı: TÜBİTAK

In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric.

Anahtar Kelimeler:

Almost complex structure, holomorphic tensor field, Kähler-Norden metric, Killing vector field, Riemannian curvature tensors

In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric.

Keywords:

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.