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Notes on the tangent bundle with deformed complete lift metric

In the context of Riemannian geometry, the tangent bundle TM of a Riemannian manifold (M, g) was classically equipped with the Sasaki metric gS , which was introduced in 1958 by Sasaki. In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 1038 – 1049 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1402-30 Notes on the tangent bundle with deformed complete lift metric 2,∗ ¨ Aydın GEZER1 , Mustafa OZKAN Department of Mathematics, Faculty of Science, Atat¨ urk University, Erzurum, Turkey 2 Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, Ankara, Turkey 1 Received: 10.02.2014 • Accepted: 16.07.2014 • Published Online: 24.10.2014 • Printed: 21.11.2014 Abstract: In this paper, our aim is to study some properties of the tangent bundle with a deformed complete lift metric. Key words: Almost complex structure, holomorphic tensor field, K¨ ahler-Norden metric, Killing vector field, Riemannian curvature tensors 1. Introduction In the context of Riemannian geometry, the tangent bundle T M of a Riemannian manifold (M, g) was classically equipped with the Sasaki metric gS , which was introduced in 1958 by Sasaki [14]. The study of the relationship between the geometry of a manifold (M, g) and that of its tangent bundle T M equipped with the Sasaki metric gS has shown some kinds of rigidity (see [7, 9]). Other (classes of) metrics defined by the various kinds of classical lifts of the metric g from M to T M were defined in [19], and then geometers obtained interesting results related to these metrics involving the different aspects and concepts of differential geometry. If (M, J, g) is an almost Hermitian manifold, its tangent bundle T M is also an almost Hermitian manifold with almost Hermitian structure (HJ, gS ), where HJ is the horizontal lift of J [19]. In [20] (see also [21, 22]), Zayatuev studied the almost Hermitian structure on T M given by (HJ, ge) , where the metric ge is defined by (H H ) X, Y (H V ) ge X, Y ( ) ge V X,V Y ge = f g (X, Y ) , ( ) = ge V X,H Y = 0, = g (X, Y ) for all vector fields X and Y on M , and f > 0, f ∈ C ∞ (M ). For f = 1 , it follows that ge