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The second-order tangent bundle with deformed $2$ nd lift metric

Abdullah Magden

Department of Mathematics, Ataturk University, Erzurum 25240, Turkey

E-mail Address: amagden@atauni.edu.tr

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Kubra Karaca

Department of Mathematics, Ataturk University, Erzurum 25240, Turkey

E-mail Address: kubrakaraca91@gmail.com

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, and

Aydin Gezer

Department of Mathematics, Ataturk University, Erzurum 25240, Turkey

E-mail Address: aydingzr@gmail.com

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https://doi.org/10.1142/S0219887819500622Cited by:0 (Source: Crossref)

Abstract

Let $(M, g)$ be a pseudo-Riemannian manifold and $T^{2} M$ be its second-order tangent bundle equipped with the deformed $2$ nd lift metric $\bar{g}$ which is obtained from the $2$ nd lift metric by deforming the horizontal part with a symmetric $(0, 2)$ -tensor field $c$ . In the present paper, we first compute the Levi-Civita connection and its Riemannian curvature tensor field of $(T^{2} M, \bar{g})$ . We give necessary and sufficient conditions for $(T^{2} M, \bar{g})$ to be semi-symmetric. Secondly, we show that $(T^{2} M, \bar{g})$ is a plural-holomorphic $B$ -manifold with the natural integrable nilpotent structure. Finally, we get the conditions under which $(T^{2} M, \bar{g})$ with the $2$ nd lift of an almost complex structure is an anti-Kähler manifold.

Keywords:

AMSC: 53C07, 53C15, 53C35

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