Some results on Riemannian $g$-natural metrics generated by classical lifts on the tangent bundle

L. Bilen ; A. Gezer

Eurasian mathematical journal, Tome 8 (2017) no. 4, pp. 18-34.

Résumé

Let $(M, g)$ be an $n$-dimensional Riemannian manifold and $TM$ its tangent bundle equipped with Riemannian $g$-natural metrics which are linear combinations of the three classical lifts of the base metric with constant coefficients. The purpose of the present paper is three-fold. Firstly, to study conditions for the tangent bundle $TM$ to be locally conformally flat. Secondly, to define a metric connection on the tangent bundle $TM$ with respect to the Riemannian $g$-natural metric and study some its properties. Finally, to classify affine Killing and Killing vector fields. on the tangent bundle $TM$.

Bibliographie

[1] M.T.K. Abbassi, “Note on the classification theorems of g-natural metrics on the tangent bundle of a Riemannian manifold (M, g)”, Comment. Math. Univ. Carolin., 45:4 (2004), 591–596 | MR

[2] M.T.K. Abbassi, M. Sarih, “On some hereditary properties of Riemannian g-natural metrics on tangent bundles of Riemannian manifolds”, Differential Geom. Appl., 22:1 (2005), 19–47 | DOI | MR

[3] M.T.K. Abbassi, M. Sarih, “On natural metrics on tangent bundles of Riemannian manifolds”, Arch. Math., 41 (2005), 71–92 | MR

[4] M.T.K. Abbassi, M. Sarih, “On Riemannian $g$-natural metrics of the form $a^Sg+b^Hg+cVg$ on the tangent bundle of a Riemannian manifold (M, g)”, Mediterr. J. Math., 2:1 (2005), 19–43 | DOI | MR

[5] C. Dicu, C. Udriste, “Properties of the pseudo-Riemannian manifold ($TM$, $G=^Cg-^Dg-^Vg$), constructed over the Riemannian manifold (M, g)”, Mathematica (Cluj), 21(44):1 (1979), 33–41 | MR

[6] A. Gezer, O. Tarakci, A.A. Salimov, “On the geometry of tangent bundles with the metric II + III”, Ann. Polon. Math., 97:1 (2010), 73–85 | DOI | MR

[7] S. Gudmundsson, E. Kappos, “On the Geometry of the Tangent Bundles”, Expo. Math., 20 (2002), 1–41 | DOI | MR

[8] I. Hasegawa, K. Yamauchi, “Infinitesimal conformal transformations on tangent bundles with the lift metric I+II”, Sci. Math. Jpn., 57:1 (2003), 129–137 | MR

[9] I. Kolar, P. W. Michor, J. Slovak, Natural operations in differential geometry, Springer-Verlang, Berlin, 1993 | MR

[10] O. Kowalski, M. Sekizawa, “Natural transformation of Riemannian metrics on manifolds to metrics on tangent bundles-a classification”, Bull. Tokyo Gakugei Univ., 40:4 (1988), 1–29 | MR

[11] V. Oproiu, “Some new geometric structures on the tangent bundle”, Publ. Math. Debrecen, 55 (1999), 261–281 | MR

[12] V. Oproiu, “A locally symmetric Kaehler Einstein structure on the tangent bundle of a space form”, Beiträge Algebra Geom., 40:2 (1999), 363–372 | MR

[13] V. Oproiu, “A Kähler Einstein structure on the tangent bundle of a space form”, Int. J. Math. Math. Sci., 25:3 (2001), 183–195 | DOI | MR

[14] V. Oproiu, N. Papaghiuc, “Some classes of almost anti-Hermitian structures on the tangent bundle”, Mediterr. J. Math., 1:3 (2004), 269–282 | DOI | MR

[15] S. Sasaki, “On the differential geometry of tangent bundles of Riemannian manifolds”, Tohoku Math. J., 10 (1958), 338–358 | DOI | MR

[16] S. Tanno, “Infinitesimal isometries on the tangent bundles with complete lift metric”, Tensor (N.S.), 28 (1974), 139–144 | MR

[17] O. Tarakci, A. Gezer, A.A. Salimov, “On solutions of IHPT equations on tangent bundles with the metric II + III”, Math. Comput. Modelling, 50:7–8 (2009), 953–958 | DOI | MR

[18] C. Udrişte, “On the metric II +III in the tangent bundle of a Riemannian manifold”, Rev. Roumaine Math. Pures Appl., 22:9 (1977), 1309–1320 | MR

[19] K. Yamauchi, “On infinitesimal conformal transformations of the tangent bundles over Riemannian manifolds”, Ann Rep. Asahikawa. Med. Coll., 15 (1994), 1–10

[20] K. Yano, S. Ishihara, Tangent and cotangent bundles, Marcel Dekker, Inc., New York, 1973 | MR