On the rescaled Riemannian metric of Cheeger-Gromoll type on the cotangent bundle

Abstract

Let (M, g) be an n−dimensional Riemannian manifold and T ∗M be its cotangent bundle equipped with a Riemannian metric of CheegerGromoll type which rescale the horizontal part by a positive differentiable function. The main purpose of the present paper is to discuss curvature properties of T ∗M and construct almost paracomplex Norden structures on T ∗M. We investigate conditions for these structures to be para-Kähler (paraholomorphic) and quasi-para-Kähler. Also, some properties of almost paracomplex Norden structures in context of almost product Riemannian manifolds are presented. 

Keywords

• Almost paracomplex structure,connection,cotangent bundle,paraholomorphic tensor field,Riemannian metric

References

1. Agca, F. and Salimov, A. A. Some notes concerning Cheeger-Gromoll metrics, Hacet. J. Math. Stat. 42 (5), 533-549, 2013.
2. Blaga, A. M. and Crasmareanu, M. The geometry of product conjugate connections, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.), 59 (1), 73-84, 2013.
3. Cheeger, J. and Gromoll, D. On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96, 413-443, 1972.
4. Cruceanu, V. Une classe de structures géométriques sur le fibré cotangent, Tensor (N.S.) 53 Commemoration Volume I, 196-201, 1993.
5. Druta, L. S. Classes of general natural almost anti-Hermitian structures on the cotangent bundles, Mediterr. J. Math. 8 (2), 161-179, 2011.
6. Fujimoto, A. Theory of G-structures, Publ. Study Group of Geometry, 1, Tokyo Univ., Tokyo, 1972.
7. Gezer, A. and Altunbas, M. Some notes concerning Riemannian metrics of Cheeger-Gromoll type, J. Math. Anal. Appl. 396 (1), 119-132, 2012.
8. Gezer, A. and Altunbas, M. Notes on the rescaled Sasaki type metric on the cotangent bundle, Acta Math. Sci. Ser. B Engl. Ed. 34 (1), 162-174, 2014.

9. Gil-Medrano O. and Naveira, A. M. Some remarks about the Riemannian curvature operator of a Riemannian almost-product manifold, Rev. Roum. Math. Pures Appl. 30 (18), 647-658, 1985.
10. Gray, A. and Hervella, L.M. The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. IV. Ser. 123, 35-58, 1980.
11. Kowalski, O. and Sekizawa, M. On the geometry of orthonormal frame bundles, Math. Nachr. 281 (12), 1799-1809, 2008.
12. de Leon, M. and Rodrigues, P. R. Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 1989.
13. Manev, M. and Mekerov, D. On Lie groups as quasi-Kähler manifolds with Killing Norden metric, Adv. Geom. 8 (3), 343-352, 2008.
14. Musso, E. and Tricerri, F. Riemannian Metrics on Tangent Bundles, Ann. Mat. Pura. Appl. 150 (4), 1-19, 1988.
15. Naveira, A.M. A classification of Riemannian almost-product manifolds, Rend. Mat.Appl. VII. Ser. 3, 577-592, 1983.
16. Olszak, Z. On almost complex structures with Norden metrics on tangent bundles, Period. Math. Hungar. 51 (2), 59-74, 2005.
17. Oproiu, V. and Papaghiuc, N. Einstein quasi-anti-Hermitian structures on the tangent bundle, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 50 (2), 347-360, 2004.
18. Oproiu, V. and Papaghiuc, N. Classes of almost anti-Hermitian structures on the tangent bundle of a Riemannian manifold, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 50 (1), 175-190, 2004.
19. Oproiu, V. and Papaghiuc, N. Some classes of almost anti-Hermitian structures on the tangent bundle, Mediterr. J. Math. 1 (3), 269-282, 2004.
20. Papaghiuc, N. A locally symmetric complex structure with Norden metric on the tangent bundle of a space form, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 48 (1), 101-112, 2002.
21. Salimov, A. A., Iscan, M. and Etayo, F. Paraholomorphic B-manifold and its properties, Topology Appl. 154 (4), 925-933, 2007.
22. Salimov, A., Gezer, A. and Iscan, M. On para-Kähler-Norden structures on the tangent bundles, Ann. Polon. Math. 103 (3), 247-261, 2012.
23. Salimov, A. and 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. 365
24. Salimov, A. A., Iscan, M. and Akbulut, K. Notes on para-Norden-Walker 4-manifolds, Int. J. Geom. Methods Mod. Phys. 7 (8), 1331-1347, 2010.
25. Sasaki, S. On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10, 338-358, 1958.
26. Staikova, M.T. and Gribachev, K. I. Canonical connections and their conformal invariants on Riemannian almost-product manifolds, Serdica 18 (3-4), 150-161, 1992.
27. Yano, K. and Ako, M. On certain operators associated with tensor field, Kodai Math. Sem. Rep. 20, 414-436, 1968.
28. Yano, K.and Ishihara, S. Tangent and Cotangent Bundles, Marcel Dekker, Inc., New York 1973.
29. Yano, K. Differential geometry on complex and almost complex spaces, Pure and Applied Math. vol. 49, New York, Pergamon Press Book, 1965.
30. Zayatuev, B. V. On some classes of almost-Hermitian structures on the tangent bundle, Webs and Quasigroups. T.S.U. 103–106, 2002.