Research Article
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Year 2023, Volume: 72 Issue: 2, 363 - 373, 23.06.2023
https://doi.org/10.31801/cfsuasmas.1067247

Abstract

References

  • 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
  • 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
  • Aslanci, S., Kazimova, S., Salimov, A. A, Some notes concerning Riemannian extensions, Ukrainian Math. J., 62(5) (2010), 661–675.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Dryuma, V., The Riemannian extension in theory of differential equations and their application, Mat. Fiz. Anal. Geom., 10(3) (2003), 307–325.
  • 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
  • Kowalski, O., Sekizawa, M., On natural Riemann extensions, Publ. Math. Debr., 78(3–4) (2011), 709-721. DOI: 10.5486/PMD.2011.4992
  • 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
  • Ocak, F., Notes about a new metric on the cotangent bundle, Int. Electron. J. Geom., 12(2) (2019), 241–249.
  • Ocak, F., Some properties of the Riemannian extensions, Konuralp J. of Math., 7(2) (2019), 359–362.
  • Ocak, F., Some notes on Riemannian extensions, Balkan J. Geom. Appl., 24(1) (2019), 45–50.
  • 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.
  • Patterson, E. M., Walker, A. G., Riemannian extensions, Quant. Jour. Math., 3 (1952), 19–28.
  • 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
  • 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.
  • Yano, K., Ishihara, S., Tangent and Cotangent Bundles, Mercel Dekker, Inc., New York, 1973.

Notes on some properties of the natural Riemann extension

Year 2023, Volume: 72 Issue: 2, 363 - 373, 23.06.2023
https://doi.org/10.31801/cfsuasmas.1067247

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.

References

  • 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
  • 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
  • Aslanci, S., Kazimova, S., Salimov, A. A, Some notes concerning Riemannian extensions, Ukrainian Math. J., 62(5) (2010), 661–675.
  • 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
  • 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
  • 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
  • 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
  • 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
  • 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
  • Dryuma, V., The Riemannian extension in theory of differential equations and their application, Mat. Fiz. Anal. Geom., 10(3) (2003), 307–325.
  • 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
  • Kowalski, O., Sekizawa, M., On natural Riemann extensions, Publ. Math. Debr., 78(3–4) (2011), 709-721. DOI: 10.5486/PMD.2011.4992
  • 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
  • Ocak, F., Notes about a new metric on the cotangent bundle, Int. Electron. J. Geom., 12(2) (2019), 241–249.
  • Ocak, F., Some properties of the Riemannian extensions, Konuralp J. of Math., 7(2) (2019), 359–362.
  • Ocak, F., Some notes on Riemannian extensions, Balkan J. Geom. Appl., 24(1) (2019), 45–50.
  • 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.
  • Patterson, E. M., Walker, A. G., Riemannian extensions, Quant. Jour. Math., 3 (1952), 19–28.
  • 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
  • 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.
  • Yano, K., Ishihara, S., Tangent and Cotangent Bundles, Mercel Dekker, Inc., New York, 1973.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Filiz Ocak 0000-0003-4157-6404

Publication Date June 23, 2023
Submission Date February 2, 2022
Acceptance Date November 27, 2022
Published in Issue Year 2023 Volume: 72 Issue: 2

Cite

APA Ocak, F. (2023). Notes on some properties of the natural Riemann extension. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(2), 363-373. https://doi.org/10.31801/cfsuasmas.1067247
AMA Ocak F. Notes on some properties of the natural Riemann extension. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. June 2023;72(2):363-373. doi:10.31801/cfsuasmas.1067247
Chicago Ocak, Filiz. “Notes on Some Properties of the Natural Riemann Extension”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, no. 2 (June 2023): 363-73. https://doi.org/10.31801/cfsuasmas.1067247.
EndNote Ocak F (June 1, 2023) Notes on some properties of the natural Riemann extension. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 2 363–373.
IEEE F. Ocak, “Notes on some properties of the natural Riemann extension”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 72, no. 2, pp. 363–373, 2023, doi: 10.31801/cfsuasmas.1067247.
ISNAD Ocak, Filiz. “Notes on Some Properties of the Natural Riemann Extension”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/2 (June 2023), 363-373. https://doi.org/10.31801/cfsuasmas.1067247.
JAMA Ocak F. Notes on some properties of the natural Riemann extension. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:363–373.
MLA Ocak, Filiz. “Notes on Some Properties of the Natural Riemann Extension”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 72, no. 2, 2023, pp. 363-7, doi:10.31801/cfsuasmas.1067247.
Vancouver Ocak F. Notes on some properties of the natural Riemann extension. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(2):363-7.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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