On the geometry of the tangent bundle with vertical rescaled metric. (English) Zbl 1487.53036
Summary: 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-Cività 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.
MSC:
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
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. · Zbl 1097.53013 |
| [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. · Zbl 1068.53016 |
| [3] | Abbassi, M., T., K., Sarih, M., On natural metrics on tangent bundles of Riemannian manifolds, Arch. Math. 41(2005), 71-92. · Zbl 1114.53015 |
| [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. · Zbl 1175.53024 |
| [5] | Dombrowski, P., On the Geometry of the Tangent Bundle, J. Reine Angew Math. 210(1962), 73-88. · Zbl 0105.16002 |
| [6] | Cheeger, J., Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. of Math, 96(1972), 413-443. · Zbl 0246.53049 |
| [7] | García-Río, D., Kupeli, N., Semi-Riemannian Maps and Their Applications, Mathematics and Its Applications, Springer science media, B.V.8 2010. · Zbl 0924.53003 |
| [8] | Gezer, A., On the tangent bundle with deformed Sasaki metric, International Electronic Journal of Geometry, 6(2013), 19-31. · Zbl 1308.53046 |
| [9] | Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundle with the Cheeger-Gromoll metric, Tokyo J. Math. 25(2002), 75-83. · Zbl 1019.53017 |
| [10] | Gudmundsson, S., Kappos, E., On the Geometry of the Tangent Bundles, Expo. Math. 20(2002), 1-41. · Zbl 1007.53027 |
| [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. · Zbl 1106.53011 |
| [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. · Zbl 0222.53044 |
| [13] | Musso, E., Tricerri, F., Riemannian metric on tangent bundle, Ann. Math. Pura. Appl. 150(1988), 1-19. · Zbl 0658.53045 |
| [14] | O’Neill, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983. · Zbl 0531.53051 |
| [15] | Oproiu, V., Some new geometric structures on the tangent bundles. Publ Math. Debrecen, 55(1999) 261-281. · Zbl 0992.53053 |
| [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. · Zbl 1011.53048 |
| [17] | Sasaki, S., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10(1958) 338-358. · Zbl 0086.15003 |
| [18] | Sekizawa, M., Curvatures of Tangent Bundles with Cheeger-Gromoll metric, Tokyo J. Math. 14(1991) 407-417. · Zbl 0768.53020 |
| [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. · Zbl 0262.53024 |
| [21] | Zayatuev, B. V., On geometry of tangent Hermitian surface, Webs and Quasigroups. T.S.U. (1995) 139-143. · Zbl 0844.53026 |
| [22] | Zayatuev, B. V., On some classes of almost-Hermitian structures on the tangent bundle, Webs and Quasigroups. T.S.U. (2002) 103-106. · Zbl 1081.53512 |
| [23] | Zhong, H. H., Lei, S., Geometry of tangent bundle with Cheeger-Gromoll type metric, Math. Anal. Appl. 402(2013) 493-504. · Zbl 1263.53017 |
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