On the Geometry of the Tangent Bundle With Vertical Rescaled Metric

Abstract

Let (M,g) be a n-dimensional smooth Riemannian manifold. In the present paper, we introduce a new class of natural metrics denoted by G^{f} and called the vertical rescaled metric on the tangent bundle TM. We calculate its Levi-Civita connection and Riemannian curvature tensor. We study the geometry of (TM,G^{f}) and several important results are obtained on curvature, Einstein structure, scalar and sectional curvatures

Keywords

• Tangent bundle,vertical rescaled metric,Einstein structure,curvature

References

1. Abbassi, M., T., K., Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M; g), Comment. Math. Univ. Carolin. 45(2004), 591-596.
2. Abbassi, M., T., K., Sarih, M., On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds, Differential Geom. Appl. 22(2005), 19-47.
3. Abbassi, M., T., K., Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. 41(2005), 71-92.
4. Dida, M., H., Hathout, F., Djaa, M., On the Geometry of the Second Order Tangent Bundle with the Diagonal lift Metric, Int. Journal of Math. Analysis. 3(2009), 443-456.
5. Dombrowski, P., On the Geometry of the Tangent Bundle, J. Reine Angew Math. 210(1962), 73-88.
6. Cheeger, J., Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math, 96(1972), 413-443.
7. García-Río, D., Kupeli, N., Semi-Riemannian Maps and Their Applications, Mathematics and Its Applications, Springer science media, B.V.8 2010.
8. Gezer, A., On the tangent bundle with deformed Sasaki metric, International Electronic Journal of Geometry, 6(2013), 19-31.

9. Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25(2002), 75-83.
10. Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundles, Expo. Math. 20(2002), 1-41.
11. Hathout, F. Dida, H. M., Diagonal lift in the tangent bundle of order two and its applications, Turk. J. Math 30(2006), 373-384.
12. Kowalski, O., Curvature of the induced Riemannian metric of the tangent bundle of a Riemannian manifold, J. Reine Angew.math. 250(1971), 124-129.
13. Musso, E., Tricerri, F., Riemannian metric on tangent bundle, Ann. Math. Pura. Appl. 150(1988), 1-19.
14. O'Neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983.
15. Oproiu, V., Some new geometric structures on the tangent bundles. Publ Math. Debrecen, 55(1999) 261-281.
16. Oproiu, V., Papaghiuc, N., On the geometry of tangent bundle of a (pseudo-) Riemannian manifold, An Stiint Univ Al I Cuza Iasi Mat 44(1998) 67-83.
17. Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10(1958) 338-358.
18. Sekizawa, M., Curvatures of Tangent Bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14(1991) 407-417.
19. Wang, J., Wang, Y., On the geometry of tangent bundles with the rescaled metric, arXiv:1104.5584v1.
20. Yano, K., Ishihara, S., Tangent and cotangent bundles, Marcel Dekker, Inc., New York 1973.
21. Zayatuev, B. V., On geometry of tangent Hermitian surface, Webs and Quasigroups. T.S.U. (1995) 139-143.
22. Zayatuev, B. V., On some classes of almost-Hermitian structures on the tangent bundle, Webs and Quasigroups. T.S.U. (2002) 103--106.
23. Zhong, H. H., Lei, S., Geometry of tangent bundle with Cheeger--Gromoll type metric, Math. Anal. Appl. 402(2013) 493-504.