On the cotangent bundle with vertical modified riemannian extensions
DOI:
https://doi.org/10.5269/bspm.64108Abstract
Let $M$ be an n-dimensional differentiable manifold with a torsion-free linear connection $\nabla $ which induces on its cotangent bundle ${T^*}M$. The main purpose of the present paper is to study some properties of the vertical modified Riemannian extension on ${T^*}M$ which is given as a new metric in [17]. At first, we investigate a metric connection with torsion on ${T^*}M$. And then, we present the holomorphy properties with respect to a compatible almost complex structure. urthermore, we study locally decomposable Golden pseudo-Riemannian structures on the cotangent bundle endowed with vertical modified Riemannian extension.References
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2. Aslanci, S., Kazimova , S., Salimov, AA., Some notes concerning Riemannian extensions, Ukrainian Math. J. 62(5), 661-675, (2010).
3. Bejan, C.L., Eken, S，., A characterization of the Riemann extension in terms of harmonicity, Czech. Math. J. 67(1), 197-206, (2017).
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10. Gezer, A., Altunbas, M., Notes on the rescaled Sasaki type metric on the cotangent bundle, Acta Math. Sci. 34B(1), 162-174, (2014).
11. Gezer, A., Bilen, L., Cakmak, A., Properties of Modified Riemannian Extensions, Zh. Mat. Fiz. Anal. Geom. 11(2), 159-173, (2015).
12. Gezer, A., Cengiz, N., Salimov, A., On integrability of Golden Riemannnian structures, Turk. J. Math. 37, 693-703, (2013).
13. Hayden, H.A., Sub-spaces of a space with torsion, Proc. Lond. Math. Soc. 34, 27-50, (1932).
14. Hretcanu, C.E., Crasmareanu, M., Applications of the golden ratio on Riemannian manifolds, Turk. J. Math. 33, 179-191, (2009).
15. Iscan, M., Salimov, A.A., On Kahler Norden manifolds, Proc. Indian Acad. Sci. (Math. Sci.) 119(1), 71-80, (2009).
16. Norden, A.P., On a certain class of four-dimensional A-spaces, Izv. Vuzov. Mat. 4, 145-157, (1960).
17. Ocak, F., Notes about a new metric on the cotangent bundle, Int. Electron. J. Geom. 12(2), 241-249, (2019).
18. Ocak, F., Some properties of the Riemannian extensions, Konuralp J. of Math. 7(2), 359-362, (2019).
19. Ocak, F., Some Notes on Riemannian Extensions, Balkan J. Geom. Appl. 24(1), 45-50, (2019).
20. Ocak, F., Kazimova, S., On a new metric in the cotangent bundle, Transactions of NAS of Azerbaijan Series of Physical-Technical and Mathematical Sciences 38(1), 128-138, (2018).
21. Ozkan, M., Prolongations of Golden structures to tangent bundles, Diff. Geom. Dyn. Syst. 16, 227-238, (2014).
22. Patterson, E.M., Walker, A.G., Riemann Extensions, Quart. J. Math. Oxford Ser. 3, 19-28, (1952).
23. Salimov, A.A., On operators associated with tensor fields, J. Geom. 99(1-2), 107-145, (2010).
24. Salimov, A., Cakan, R., On deformed Riemannian extensions associated with twin Norden metrics, Chinese Annals of Math. Ser.B. 36, 345-354, (2015).
25. Salimov, A.A., Iscan, M., Etayo, F., Paraholomorphic B-manifold and its properties, Topol. Appl. 154, 925-933, (2007).
26. Tachibana, S., Analytic tensor and its generalization, Tohoku Math. J. 12, 208– 221, (1960).
27. Vishnevskii, V.V., Integrable affinor structures and their plural interpretations, J. of Math. Sci. 108(2), 151-187, (2002).
28. Yano, K., Ako, M., On certain operators associated with tensor fields, Kodai Math. Sem. Rep. 20, 414-436, (1968).
29. Yano, K., Ishihara, S., Tangent and Cotangent Bundles, New York, USA, Marcel Dekker, (1973).
2. Aslanci, S., Kazimova , S., Salimov, AA., Some notes concerning Riemannian extensions, Ukrainian Math. J. 62(5), 661-675, (2010).
3. Bejan, C.L., Eken, S，., A characterization of the Riemann extension in terms of harmonicity, Czech. Math. J. 67(1), 197-206, (2017).
4. Bejan, C.L., Kowalski, O., On some differential operators on natural Riemann extensions, Ann. Glob. Anal. Geom. 48, 171-180, (2015).
5. Bejan, C.L., Meri，c, S，.E., Kılı，c, E., Einstein Metrics Induced by Natural Riemann Extensions, Adv. Appl. Clifford Algebras 27(3), 2333-2343, (2017).
6. Bilen, L., Gezer, A., On metric connections with torsion on the cotangent bundle with modified Riemannian extension, J. Geom. 109(6), 1-17, (2018).
7. Crasmareanu, M., Hretcanu, C.E., Golden differential geometry, Chaos Solitons and Fractals 38, 1229-1238, (2008).
8. Calvino-Louzao, E., Garcıa-Rıo, E., Gilkey, P., Vazquez-Lorenzo, A., The Geometry of Modified Riemannian Extensions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465(2107), 2023-2040, (2009).
9. Ganchev, G. T., Borisov, A.V., Note on the almost complex manifolds with a Norden metric, C. R. Acad. Bulg. Sci. 39, 31-34, (1986).
10. Gezer, A., Altunbas, M., Notes on the rescaled Sasaki type metric on the cotangent bundle, Acta Math. Sci. 34B(1), 162-174, (2014).
11. Gezer, A., Bilen, L., Cakmak, A., Properties of Modified Riemannian Extensions, Zh. Mat. Fiz. Anal. Geom. 11(2), 159-173, (2015).
12. Gezer, A., Cengiz, N., Salimov, A., On integrability of Golden Riemannnian structures, Turk. J. Math. 37, 693-703, (2013).
13. Hayden, H.A., Sub-spaces of a space with torsion, Proc. Lond. Math. Soc. 34, 27-50, (1932).
14. Hretcanu, C.E., Crasmareanu, M., Applications of the golden ratio on Riemannian manifolds, Turk. J. Math. 33, 179-191, (2009).
15. Iscan, M., Salimov, A.A., On Kahler Norden manifolds, Proc. Indian Acad. Sci. (Math. Sci.) 119(1), 71-80, (2009).
16. Norden, A.P., On a certain class of four-dimensional A-spaces, Izv. Vuzov. Mat. 4, 145-157, (1960).
17. Ocak, F., Notes about a new metric on the cotangent bundle, Int. Electron. J. Geom. 12(2), 241-249, (2019).
18. Ocak, F., Some properties of the Riemannian extensions, Konuralp J. of Math. 7(2), 359-362, (2019).
19. Ocak, F., Some Notes on Riemannian Extensions, Balkan J. Geom. Appl. 24(1), 45-50, (2019).
20. Ocak, F., Kazimova, S., On a new metric in the cotangent bundle, Transactions of NAS of Azerbaijan Series of Physical-Technical and Mathematical Sciences 38(1), 128-138, (2018).
21. Ozkan, M., Prolongations of Golden structures to tangent bundles, Diff. Geom. Dyn. Syst. 16, 227-238, (2014).
22. Patterson, E.M., Walker, A.G., Riemann Extensions, Quart. J. Math. Oxford Ser. 3, 19-28, (1952).
23. Salimov, A.A., On operators associated with tensor fields, J. Geom. 99(1-2), 107-145, (2010).
24. Salimov, A., Cakan, R., On deformed Riemannian extensions associated with twin Norden metrics, Chinese Annals of Math. Ser.B. 36, 345-354, (2015).
25. Salimov, A.A., Iscan, M., Etayo, F., Paraholomorphic B-manifold and its properties, Topol. Appl. 154, 925-933, (2007).
26. Tachibana, S., Analytic tensor and its generalization, Tohoku Math. J. 12, 208– 221, (1960).
27. Vishnevskii, V.V., Integrable affinor structures and their plural interpretations, J. of Math. Sci. 108(2), 151-187, (2002).
28. Yano, K., Ako, M., On certain operators associated with tensor fields, Kodai Math. Sem. Rep. 20, 414-436, (1968).
29. Yano, K., Ishihara, S., Tangent and Cotangent Bundles, New York, USA, Marcel Dekker, (1973).
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2024-05-08
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