Notes About a New Metric on the Cotangent Bundle

Year 2019, Volume: 12 Issue: 2, 241 - 249, 03.10.2019

Filiz Ocak

https://doi.org/10.36890/iejg.542783

Cited By: 6

Abstract

In this article, we construct a new metric $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}  \over G}  = {}^R\nabla  + \sum\limits_{i,j = 1}^m {a^{ji}} \delta {p_j}\delta {p_i}$  in the cotangent  bundle, where ${}^R\nabla $ is the  Riemannian extension and  $ a^{ji}$ is a symmetric (2,0)-tensor field on a differentiable manifold.

Keywords

Cotangent bundle, Riemannian extension, geodesic, para-Nordenian metric

References

• \bibitem{1} Aslanci, S., Cakan, R., On a cotangent bundle with deformed Riemannian extension, \emph{Mediterr. J. Math.} 11 (2014), 1251-1260.
• \bibitem{2} Aslanci, S., Kazimova, S., Salimov, A.A., Some Remarks Concerning Riemannian Extensions, \emph{Ukrainian. Math. J.} 62, (2010), 661-675.
• \bibitem{3}Bejan, C.L., Eken, \c{S}., A characterization of the Riemann extension in terms of harmonicity, \emph{Czech. Math. J.} 67, (2017), 197-206.
• \bibitem{4}Bejan, C.L., Meri\c{c}, \c{S}. E., K{\i}l{\i}\c{c}, E., Einstein Metrics Induced by Natural Riemann Extensions, \emph{Adv. Appl. Clifford Algebras.} 27, (2017), 2333-2343.
• \bibitem{5}Calvi\~{n}o-Louzao, E., Garc\'{i}a-R\'{i}o, E., Gilkey, P., V\'{a}zquez-Lorenzo A., The Geometry of Modified Riemannian Extensions, \emph{Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.} 465, (2009), 2023-2040.
• \bibitem{6} Cruceanu, V., Fortuny, P., Gadea, M., A survey on paracomplex Geometry, \emph{Rocky Mountain J. Math.} 26, (1995), 83-115.
• \bibitem{7} Dryuma, V., The Riemann Extensions in Theory of Differential Equations and their Applications, \emph{Mat. Fiz. Anal. Geom.} 10, (2003), 307-325.
• \bibitem{8} Gezer, A., Bilen, L., Cakmak, A., Properties of Modified Riemannian Extensions,\emph{ Zh. Mat. Fiz. Anal. Geom.} 11, (2015), 159-173.
• \bibitem{9} Kruckovic, GI., Hypercomplex structures on manifolds I. \emph{Trudy. Sem. Vektor Tenzor Anal.} 16, (1972), 174-201(in Russian).
• \bibitem{10} Ocak, F., Kazimova, S., On a new metric in the cotangent bundle, \emph{Transactions of NAS of Azerbaijan Series of Physical-Technical and Mathematical Sciences.} 38, (2018), 128-138.
• \bibitem{11} Patterson, E.M., Walker, A.G., Riemann Extensions, \emph{Quart. J. Math. Oxford Ser.} 3, (1952), 19–28.
• \bibitem{12}Salimov, A., Tensor Operators and Their Applications, Nova Science Publishers, New York, USA, 2012.
• \bibitem{13} Salimov, A., Cakan, R., On deformed Riemannian extensions associated with twin Norden metrics, \emph{Chinese Annals of Mathematics Series B. } 36, (2015), 345-354.
• \bibitem{14} Salimov, A.A., Iscan, M., Akbulut, K., Notes on para-Norden–Walker 4-manifolds, \emph{International Journal of Geometric Methods in Modern Physics.} 7, (2010), 1331-1347.
• \bibitem{15} Salimov, A. A., Iscan, M. and Etayo, F., Paraholomorphic B-manifold and its properties, \emph{Topology Appl.} 154,(2007), 925-933.
• \bibitem{16} Yano, K., Ako, M., On certain operators associated with tensor fields, \emph{Kodai Math. Sem. Rep.} 20, (1968), 414-436.
• \bibitem{17} Yano, K. and Ishihara, S., Tangent and Cotangent Bundles, Pure and Applied Mathematics, 16, Marcel Dekker, Inc., New York, 1973.