Geodesics of Twisted-Sasaki Metric

Year 2021, Volume: 9 Issue: 3, 124 - 132, 30.09.2021

Zagane Abderrahım

https://doi.org/10.36753/mathenot.710119

Abstract

The main purpose of the paper is to investigate geodesics on the tangent bundle with respect to the twisted-Sasaki metric. We establish a necessary and sufficient conditions under which a curve be a geodesic respect. Afterward, we also construct some examples of geodesics.

Keywords

Tangent Bundle, horizontal lift, vertical lift, twisted-Sasaki metric, geodesics

References

• [1] Abbassi, M. T. K., Sarih, M.: On Natural Metrics on Tangent Bundles of Riemannian Manifolds. Arch. Math. (Brno) 41, 71-92 (2005).
• [2] Cengiz, N., Salimov, A. A.: Diagonal lift in the tensor bundle and its applications. Appl. Math. Comput. 142(2-3), 309-319 (2003).
• [3] Cheeger, J., Gromoll, D.: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. (2) 96, 413-443 (1972). https://doi.org/10.2307/1970819
• [4] Djaa, M., Gancarzewicz, J.: The geometry of tangent bundles of order r. boletin Academia Galega de Ciencias 4, 147-165 (1985).
• [5] Djaa, N. E. H., Boulal, A., Zagane, A.: Generalized Warped Product Manifolds And Biharmonic Maps. Acta Math. Univ. Comenian. (N.S.) 81 (2), 283-298 (2012).
• [6] Dombrowski, P.: On the Geometry of the Tangent Bundle. J. Reine Angew. Math. 210, 73-88 (1962). https: //doi.org/10.1515/crll.1962.210.73
• [7] Gezer, A.: On the Tangent Bundle With Deformed Sasaki Metric. Int. Electron. J. Geom. 6 (2), 19-31 (2013).
• [8] Gudmundsson, S., Kappos, E.:On the Geometry of the Tangent Bundle with the Cheeger-Gromoll Metric. Tokyo J. Math. 25 (1), 75-83 (2002).
• [9] Jian, W., Yong, W.: On the Geometry of Tangent Bundles with the Rescaled Metric. arXiv:1104.5584v1 [math.DG] 29 Apr 2011.
• [10] Kada Ben Otmane, R., Zagane, A., Djaa, M.: On generalized Cheeger-Gromoll metric and harmonicity. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 69 (1), 629-645 (2020). https://doi.org/10.31801/cfsuasmas. 487296
• [11] Kowalski, O., Sekizawa, M.: On Riemannian Geometry Of Tangent Sphere Bundles With Arbitrary Constant Radius. Arch. Math. (Brno) 44, 391-401 (2008).
• [12] Latti, F., Djaa, M., Zagane, A.: Mus-Sasaki Metric and Harmonicity. Mathematical Sciences and Applications E-Notes 6 (1) 29-36 (2018). https://doi.org/10.36753/mathenot.421753
• [13] Musso,E.,Tricerri,F.:RiemannianMetricsonTangentBundles.Ann.Mat.Pura.Appl.150,1-19(1988).https: //doi.org/10.1007/BF01761461
• [14] Salimov, A. A., Gezer, A., Akbulut, K.: Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math. 6 (2), 135-147 (2009). https://doi.org/10.1007/s00009-009-0001-z
• [15] Salimov, A. A, Gezer, A.: On the geometry of the (1, 1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B 32 (3), 369-386 (2011). https://doi.org/10.1007/s11401-011-0646-3
• [16] Salimov, A. A., Agca, F.: Some Properties of Sasakian Metrics in Cotangent Bundles. Mediterr. J. Math. 8 (2), 243-255 (2011). https://doi.org/10.1007/s00009-010-0080-x
• [17] Salimov, A. A., Kazimova, S.: Geodesics of the Cheeger-Gromoll Metric. Turkish J. Math. 33, 99-105 (2009). doi:10.3906/mat-0804-24
• [18] Sasaki,S.: OnthedifferentialgeometryoftangentbundlesofRiemannianmanifolds,II.TohokuMath.J.(2)14(2), 146-155 (1962). https://doi.org/10.2748/tmj/1178244169
• [19] Sekizawa,M.: CurvaturesoftangentbundleswithCheeger-Gromollmetric.TokyoJ.Math.14(2),407-417(1991). DOI10.3836/tjm/1270130381
• [20] Yano, K., Ishihara, S.: Tangent and tangent bundles. Marcel Dekker. INC. New York (1973).
• [21] Zagane, A., Djaa, M.: On Geodesics of Warped Sasaki Metric. Mathematical Sciences and Applications E-Notes 5 (1), 85-92 (2017). https://doi.org/10.36753/mathenot.421709
• [22] Zagane, A., Djaa, M.: Geometry of Mus-Sasaki metric. Commun. Math. 26 (2), 113-126 (2018). https://doi. org/10.2478/cm-2018-0008