Geodesics on the Cotangent Bundle with Vertical Rescaled Cheeger-Gromoll Metric

Year 2021, Volume: 3 Issue: 1, 1 - 8, 30.08.2021

Zagane Abderrahım

Abstract

In this paper, we introduce the vertical rescaled Cheeger-Gromoll metric (deformation in the vertical bundle) on the cotangent bundle T*M over a Riemannian manifold (M, g) and we investigate the Levi-Civita connection of this metric. We study the geodesics on the cotangent bundle with respect to the vertical rescaled Cheeger-Gromoll metric. Afterward, we establish the necessary and sufficient conditions under which a curve be geodesic respect to this metric. Finally, we also construct some examples of geodesics.

Keywords

cotangent bundles, vertical rescaled Cheeger-Gromoll metric, geodesics

References

• Ağca, F. (2013). g-Natural Metrics on the cotangent bundle. Int. Electron. J. Geom. 6(1), 129-146.
• Ağca, F., & Salimov, A. A. (2013). Some notes concerning Cheeger-Gromoll metrics. Hacet. J. Math. Stat. 42(5), 533-549.
• Gezer, A., & Altunbas, M. (2016). On the rescaled Riemannian metric of Cheeger Gromoll type on the cotangent bundle. Hacet. J. Math. Stat. 45(2), 355-365.
• Ocak, F. (2019). Notes about a new metric on the cotangent bundle. Int. Electron. J. Geom. 12(2), 241-249
• Patterson, E. M., & Walker, A. G. (1952). Riemannian extensions. Quart. J.Math. Oxford Ser. 3(1), 19-28.
• Salimov, A. A., & Ağca, F. (2011). Some properties of Sasakian metrics in cotangent bundles. Mediterr. J. Math. 8(2), 243-255.
• Sekizawa, M. (1987). Natural transformations of affine connections on manifolds to metrics on cotangent bundles. In: Proceedings of 14th Winter School on Abstract Analysis (Srni, 1986), Rend. Circ. Mat. Palermo, 14, 129-142.
• Yano, K., & Ishihara, S. (1973). Tangent and cotangent bundles. M. Dekker, New York.
• Zagane, A. (2020). A new class of metrics on the cotangent bundle. Bulletin of the Transilvania University of Brasov. Mathematics, Informatics, Physics. Series III, 13(1), 285-301.