Quasi-Concircular Curvature Tensor on Generalized Sasakian Space-Forms

Year 2023, Volume: 15 Issue: 2, 391 - 402, 31.12.2023

Rana Pratap Singh Yadav Bhagwat Prasad

https://doi.org/10.47000/tjmcs.1194308

Abstract

The object of the present paper is to study Quasi-concircularly flat and $\phi-$quasi-concircularly flat generalized Sasakian-space-forms. Also, we consider generalized Sasakian-space-forms satisfying the condition $P(\xi, X).\widetilde{V}=0,\ \widetilde{V}(\xi, X).P=0, \ $ and $\widetilde{V}(\xi, X).\widetilde{V}=0$ and we obtain some important results. Finally, we give an example.

Keywords

Generalized Sasakian space-froms, $\xi-$quasi-concircularly flat, $\phi-$ quasi concircularly flat, Einstein and $\eta-$Einstein manifold

References

• Alegre, P., Blair, D.E., Carrazo, A., Generalized Sasakian-space-forms, Israel J. Math., 141(2004), 157–183.
• Alegre, P., Carriazo, A., Structure on generalized Sasakian-space-forms, Differential Geometry and Its Applications, 26(2008), 656–666.
• Ahmad, M., Haseeb, A., Jun, J.B., Quasi-concircular curvature tensor on a Lorentzian β−Kenmotsu manifold, Journal of the Chungcheong Mathematical Society, 32(3)(2019).
• Blair, D.E., Lecture Notes in Mathematics, Springer Verleg, Berlin, 1976.
• Blair, D.E., Riemannian Geometry of Contact and Symplectic Manifolds, Birkhauser, 2002.
• Chaubey, S.K., Yadav, S.K., W-semisymmetric generalized Sasakian-space-forms, Advances in Pure and Applied Mathematics, (2018), 1–10.
• Cabrerizo, J.L., Fernandez, L.M., Fernandez, M., Zhen, G.. The structure of a class of k-contact manifolds, Acta Math. Hungar, 82(4)(1999), 331–340.
• De, U.C., Majhi, P., On the Q−curvature tensor on a generalized Sasakian-spac-form, Kragujevac Journal of Mathematics, 43(3)(2019), 333–349.
• De, U.C., Sarkar, A., Some results on generalized Sasakian-space-forms, Thai J. Math., 8(2010), 1–10.
• De, U.C., Yildiz, A., Certain curvature conditions on generalized Sasakian-space-forms, Quaest. Math., 38(4)(2015), 495–504.
• Hui, S.K., Debabrata, C., Generalized Sasakian-space-forms and Ricci almost solitons with a conformal killing vector field, NTMSCI, 4(3) (2016), 263–269 .
• Kim, U.K., Conformally flat generalized Sasakian-space-forms and locally symmetric generalized Sasakian-space-forms, Note Mat., 26(2006), 55–67.
• Majhi, P., De, U.C., The Structure of a Class of Generalized Sasakian-Space-Forms, Extracta Mathematicae, 27(2)(2012), 301–308.
• Mantica, C.A., Suh, Y.J., Pseudo−Q−symmetric Riemannian manifolds, Int. J. Geo. Methods Mod. Phys., 10(5)(2013).
• Narain, D., Prakash, A., Prasad, B., Quasi concircular curvature tensor on an Lorentzian para-Sasakian manifold, Bull. Cal. Math,. Soc., 101(4)(2009), 387–394.
• Nagaraja, H.G., Somashekhara, G., Shashidhar, S., On generalized Sasakian space-froms, ISRN Geometry, 2012(2012).
• Özgür, C., ϕ−Conformally flat LP-Sasakian manifold, Rad. Mat., 12(1)(2003), 99–106.
• Öztürk, H., Some curvature conditions on α-cosymplectic manifolds, Mathematics and Statistics, 1(4)(2013), 188–195.
• Prasad, B., Mourya, A., Quasi concircular curvature tensor, News Bull. Cal. Math. Soc., 30(1-3) (2007), 5–6.
• Prasad, B., Yadav, R.P.S., On Semi-generalized Recurrent LP-Sasakian Manifold, J. Nat. Acad. Math., 31(2017), 57–69.
• Shukla, N.C.V., Shah, R.J. , Generalized Sasakian space form with concircular curvature tensor, J. Rajasthan Acad. Phy. Sci., 10(1)(2011).
• Sular, S., Özgür, C., Generalized Sasakian space-forms with semi-symmetric non-metric connection, Proceeding of the Estonian Academy of Sciences, 60(4)(2011), 251–257.
• Shah, R.Jung, Genralized Sasakian-space forms with D−Conformal curvature tensor, Kathmandu University Journal of Science, Engineering and Technology, 8(2)(2012), 48–56.
• Sarkar, A., Akbar, A., Generalized Sasakian space-forms with projective curvature tensor, Demonstratio Mathematica, 43(3)(2014), 725–735.
• Singh, A., Kishor, S., Generalized recurrent and genaralized Ricci recurent generalized Sasakian space forms, Palestine Journal of Mathematics, 9(2)(2020), 866–873.
• Singh, J.P., Lalmalsawma, C., Symmetries of Sasakian generalized Sasakian-space-form admitting generalized Tanaka-Webster connection, Tamkang Journal of Mathematics, 52(2)(2021), 229–240.
• Singh,G., Prasad, B., Some properties of α−cosymplectic manifolds, Journal of Progressive Science, 10(01&02)(2019), 47–53.
• Venkatesha, B., Shanmukha, W2-Curvature tensor on generalized Sasakian space forms, CUBO A Mathematical Journal, 20(01)(2018), 17–29.
• Yano, K., Concircular geometry I. Concircular transformation, Proc. Imp. Acad. Tokyo, 16(1940), 195–200.
• Yano, K., Kon, M., Structures on Manifolds, Series in Pure Mathematis, World Scientific Publ. Co., Singapore, 1984.
• Zhen, G., On conformal symmetric K-contact manifolds, Chinese Quart. J. Math., 7(1992), 5–10.
• Zhen, G., Carbrerizo, J.L., Fernandez, L.M., Fernandez, M., On ξ−conformally flat contact metric manifolds, Indian J. Pure Appl. Math., 28(6)(1997), 725–734.