Notes on some properties of the natural Riemann extension

Abstract

Let $(M,\nabla)$ be an $n$-dimensional differentiable manifold with a torsion-free linear connection and $T^{*}M$ its cotangent bundle. In this context we study some properties of the natural Riemann extension (M. Sekizawa (1987), O. Kowalski and M. Sekizawa (2011)) on the cotangent bundle $T^{*}M$. First, we give an alternative definition of the natural Riemann extension with respect to horizontal and vertical lifts. Secondly, we investigate metric connection for the natural Riemann extension. Finally, we present geodesics on the cotangent bundle $T^{*}M$ endowed with the natural Riemann extension.

Keywords

• Vertical and horizontal lift
• adapted frame
• geodesics
• natural Riemann extension
• cotangent bundle

References

1. Abbasi, M. T. K., Amri, N., Bejan, C. L., Conformal vector fields and Ricci soliton structures on natural Riemannian extensions, Mediterr. J. Math., 18(55) (2021), 1-16. https://doi.org/10.1007/s00009-020-01690-5
2. Aslanci, S., Cakan, R., On a cotangent bundle with deformed Riemannian extension, Mediterr. J. Math., 11(4) (2014), 1251–1260. DOI 10.1007/s00009-013-0337-2
3. Aslanci, S., Kazimova, S., Salimov, A. A, Some notes concerning Riemannian extensions, Ukrainian Math. J., 62(5) (2010), 661–675.
4. Bejan, C. L., Eken, S¸., A characterization of the Riemann extension in terms of harmonicity, Czech. Math. J., 67(1) (2017), 197–206. DOI: 10.21136/CMJ.2017.0459-15
5. Bejan, C. L., Kowalski, O., On some differential operators on natural Riemann extensions, Ann. Glob. Anal. Geom., 48 (2015), 171–180. DOI 10.1007/s10455-015-9463-3
6. Bejan, C. L., Nakova, G., Amost para-Hermitian and almost paracontact metric structures induced by natural Riemann extensions, Resulth Math., 74(15) (2019). https://doi.org/10.1007/s00025-018-0939-x
7. Bejan, C. L., Meriç, S¸. E., Kılıç, E., Einstein metrics induced by natural Riemann extensions, Adv. Appl. Clifford Algebras, 27(3) (2017), 2333–2343. DOI 10.1007/s00006-017-0774-2
8. Bilen, L., Gezer, A., On metric connections with torsion on the cotangent bundle with modified Riemannian extension, J. Geom. 109(6) (2018), 1–17. https://doi.org/10.1007/s00022-018-0411-9

9. Calvino-Louzao, E., Garcia-Rio, E., Gilkey, P., Vazquez-Lorenzo, A., The geometry of modified Riemannian extensions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2107) (2009), 2023-2040. https://www.jstor.org/stable/30245448
10. Dryuma, V., The Riemannian extension in theory of differential equations and their application, Mat. Fiz. Anal. Geom., 10(3) (2003), 307–325.
11. Gezer, A., Bilen, L., Cakmak, A., Properties of modified Riemannian extensions, Zh. Mat. Fiz. Anal. Geom., 11(2) (2015), 159-173. https://doi.org/10.15407/mag11.02.159
12. Kowalski, O., Sekizawa, M., On natural Riemann extensions, Publ. Math. Debr., 78(3–4) (2011), 709-721. DOI: 10.5486/PMD.2011.4992
13. Kowalski, O., Sekizawa, M., Almost Osserman structures on natural Riemann extensions, Differ. Geom. Appl., 31 (2013), 140-149. https://doi.org/10.1016/j.difgeo.2012.10.007
14. Ocak, F., Notes about a new metric on the cotangent bundle, Int. Electron. J. Geom., 12(2) (2019), 241–249.
15. Ocak, F., Some properties of the Riemannian extensions, Konuralp J. of Math., 7(2) (2019), 359–362.
16. Ocak, F., Some notes on Riemannian extensions, Balkan J. Geom. Appl., 24(1) (2019), 45–50.
17. 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) (2018), 128—138.
18. Patterson, E. M., Walker, A. G., Riemannian extensions, Quant. Jour. Math., 3 (1952), 19–28.
19. Salimov, A., Cakan, R., On deformed Riemannian extensions associated with twin Norden metrics, Chinese Annals of Math. Ser. B., 36 (2015), 345–354. DOI: 10.1007/s11401-015-0914-8
20. Sekizawa, M., Natural transformations of affine connections on manifolds to metrics on cotangent bundles, Proc. 14th Winter School. Srn´ı, Czech, 1986, Suppl. Rend. Circ. Mat. Palermo, Ser., 14(2) (1987), 129—142.
21. Yano, K., Ishihara, S., Tangent and Cotangent Bundles, Mercel Dekker, Inc., New York, 1973.