SILVER STRUCTURES ON THE RIEMANN EXTENSIONS

Abstract

In the present paper we deal with an $n-$dimensional differentiable manifold $M$ with a torsion-free linear connection $\nabla $. Here we study some properties of a silver structure on the cotangent bundle ${{T}^{*}}M$ equipped with the Riemannian extension ${{}^{R}}\nabla$ and obtain a necessary condition for which the silver semi-Riemannian manifold $\left( {{T}^{*}}M{{,}^{R}}\nabla ,S \right)$ to be a locally decomposable.

Keywords

• Cotangent bundle
• Riemannian extension
• silver structure

References

1. A. Gray, Pseudo-Riemannian Almost Product Manifolds And Submersions, J.Math. Mech. Vol.16, No.7, pp.715-737 (1967).
2. S. Aslanci, S. Kazimova, A.A. Salimov, Some Remarks Concerning Riemannian Extensions, Ukrainian Math. J., Vol.62, No.5, pp.661–675 (2010).
3. C. L. Bejan, S. Eken, A Characterization Of The Riemann Extension İn Terms Of Harmonicity, Czech. Math. J., Vol.67, No.1, pp.197-206 (2017).
4. R. Cakan Akpinar, Some Results On Silver Riemannian Structures. Turkish Journal of Mathematics and Computer Science, Vol.14, No.1, pp.91-97(2022).
5. V. Dryuma, The Riemann Extensions in Theory of Differential Equations and their Applications. Mat. Fiz., Anal., Geom., Vol.10, No.3, pp.307–325 (2003).
6. F. Ocak, Some properties of the Riemannian extensions, Konuralp J. of Math., Vol.7, No.2, pp.359–362 (2019).
7. F. Ocak, Some Notes on Riemannian Extensions, Balkan J. Geom. Appl., Vol.24, No.1, pp.45–50 (2019).
8. M. Ozkan, B. Peltek, A New Structure On Manifolds: Silver Structure, İnternational Electronic Journal of Geometry, Vol.9, No.2, pp.59-69 (2016).

9. M. Ozkan, E. Taylan, A. A. Citlak, On Lifts Of Silver Structure, Journal of Science and Arts, Vol.17, No.2, pp.223 ( 2017).
10. E.M. Patterson, A.G. Walker, Riemann Extensions. Quart. J. Math. Oxford Ser., Vol.3 , pp.19–28 (1952).
11. G. T. Pripoae, Classification of Semi-Riemannian Almost Product Structure, Proceeding of the Conference of Applied Differential Geometry-General Relativity and Workshop on Global Analysis, Differential Geometry and Lie Algebras, pp.243-251 (2002).
12. A. Salimov, Tensor Operators and Their Applications, Nova Science Publishers, New York, (2012).
13. A. A. Salimov, F. Agca, On Para-Nordenian Structures, Ann. Polon. Math, Vol.99, No.2, pp.193-200 (2010).
14. A.A. Salimov, M. Iscan, F. Etayo, Paraholomorphic B-manifold and İts Properties, Topology Appl., Vol.154, pp.925-933 (2007).
15. K. Yano, M. Ako, On Certain Operators Associated With Tensor Fields, Kodai Math. Sem. Rep., Vol.20, pp. 414-436 (1968).
16. K. Yano, S. Ishihara, Tangent and Cotangent Bundles. Marcel Dekker, İnc., New York, (1973).
17. K. Yano, M. Kon, Structures on Manifolds. Singapore: World Scientific Publishing Co., (1984).