Metallic Structures on Product Manifolds and Chen-Ricci Inequalities

Year 2024, , 660 - 678, 27.10.2024

Mustafa Gök

https://doi.org/10.36890/iejg.1458491

Abstract

In this study, we discuss metallic structures on product manifolds and derive the Chen-Ricci inequalities for remarkable submanifolds determined by the behaviour of their tangent bundles with regard to the action of the metallic structure in a locally decomposable metallic Riemannian manifold whose components are spaces of constant curvature. Moreover, the equality cases are considered in order to characterize these submanifolds.

Keywords

Metallic structure, metallic Riemannian manifold, Riemannian submanifold, real space form, Ricci curvature, $k$-Ricci curvature, Chen-Ricci inequality

References

• [1] Bejancu, A.: Geometry of CR Submanifolds. D. Reidel Publishing Company, Dordrecht (1986).
• [2] Blaga, A. M.: The geometry of golden conjugate connections. Sarajevo J. Math. 10 (23), 237-245 (2014).
• [3] Blaga, A. M., Hretcanu, C. E.: Invariant, anti-invariant and slant submanifolds of a metallic Riemannian manifold. Novi Sad J. Math. 48 (2), 55-80 (2018).
• [4] Blaga, A. M., Hretcanu, C. E.: Metallic conjugate connections. Rev. Un. Mat. Argentina 59 (1), 179-192 (2018).
• [5] Blaga, A. M., Hretcanu, C. E.: Remarks on metallic warped product manifolds. Facta Univ. Ser. Math. Inform. 33 (1), 269-277 (2018).
• [6] Chen, B. Y.: Mean curvature and shape operator of isometric immersions in real space forms. Glasgow. Math. J. 38 (1), 87-97 (1996).
• [7] Chen, B. Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimension. Glasgow Math. J. 41 (1), 33-41 (1999).
• [8] Choudhary, M. A., Blaga, A. M.: Inequalities for generalized normalized δ-Casorati curvatures of slant submanifolds in metallic Riemannian space forms. J. Geom. 111 (3), Article ID 39, 18 pages (2020).
• [9] Crâşmareanu, M. C., Hretcanu, C. E.: Golden differential geometry. Chaos Soliton. Fract. 38 (5), 1229-1238 (2008).
• [10] de Spinadel, V. W.: The family of metallic means. Vis. Math. 1 (3), (1999).
• [11] de Spinadel, V. W.: The metallic means family and multifractal spectra. Nonlinear Anal. 36 (6), 721-745 (1999).
• [12] de Spinadel, V. W.: The metallic means family and renormalization group techniques. Proc. Steklov Inst. Math. (suppl. 1), 194-209 (2000).
• [13] de Spinadel, V. W.: The metallic means family and forbidden symmetries. Int. Math. J. 2 (3), 279-288 (2002).
• [14] Eken, Ş., Gülbahar, M., Kılıç, E.: Some inequalities for Riemannian submersions. An. ¸Stiin¸t. Univ. Al. I. Cuza Ia¸si. Mat. (N.S.) 63 (3), 471-482 (2017).
• [15] Etayo, F., Santamaría, R., Upadhyay, A.: On the geometry of almost golden Riemannian manifolds. Mediterr. J. Math. 14 (5), Article ID 187, 14 pages (2017).
• [16] Gezer, A., Karaman, Ç.: On metallic Riemannian structures. Turk. J. Math. 3 (6), 954-962 (2015).
• [17] Gök, M., Keleş, S., Kılıç, E.: Schouten and Vranceanu connections on golden manifolds. Int. Electron. J. Geom. 12 (2), 169-181 (2019).
• [18] Gök, M., Keleş, S., Kılıç, E.: Some characterizations of semi-invariant submanifolds of golden Riemannian manifolds. Mathematics 7 (12), Article ID 1209, 12 pages (2019).
• [19] Gülbahar, M., Eken, Ş., Kılıç, E.: Sharp inequalities involving the Ricci curvature for Riemannian submersions. Kragujevac J. Math. 41 (2), 279-293 (2017).
• [20] Hong, S., Matsumoto, K., Tripathi, M. M.: Certain basic inequalities for submanifolds of locally conformal Kahler space forms. SUT J. Math. 41 (1), 75-94 (2005).
• [21] Hong, S., Tripathi, M. M.: On Ricci curvature of submanifolds. Int. J. Pure Appl. Math. Sci. 2 (2), 227-245 (2005).
• [22] Hretcanu, C. E., Blaga, A. M.: Submanifolds in metallic Riemannian manifolds. Differ. Geom. Dyn. Syst. 20, 83-97 (2018).
• [23] Hretcanu, C. E., Blaga, A. M.: Slant and semi-slant submanifolds in metallic Riemannian manifolds. J. Funct. Spaces 2018 (3), Article ID 2864263, 13 pages (2018).
• [24] Hretcanu, C. E., Blaga, A. M.: Hemi-slant submanifolds in metallic Riemannian manifolds. Carpathian J. Math. 35 (1), 59-68 (2019).
• [25] Hretcanu, C. E., Crâşmareanu, M. C.: On some invariant submanifolds in a Riemannian manifold with golden structure. An. Ştiint. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 53 (suppl. 1), 199-211 (2007).
• [26] Hretcanu, C. E., Crâşmareanu, M. C.: Applications of the golden ratio on Riemannian manifolds. Turk. J. Math. 33 (2), 179-191 (2009).
• [27] Hretcanu, C. E., Crâşmareanu, M. C.: Metallic structures on Riemannian manifolds. Rev. Un. Mat. Argentina 54 (2), 15-27 (2013).
• [28] Kılıç, E., Tripathi, M. M., Gülbahar, M.: Chen-Ricci inequalities for submanifolds of Riemannian and Kaehlerian product manifolds. Ann. Pol. Math. 116 (1), 37-56 (2016).
• [29] Kim, J. S., Tripathi, M. M., Choi, J.: Ricci curvature of submanifolds in locally conformal almost cosyplectic manifolds. Indian J. Pure Appl. Math. 35 (3), 259-271 (2004).
• [30] Mihai, A.: Inequalities on the Ricci curvature. J. Math. Inequal. 9 (3), 811-822 (2015).
• [31] Mihai, I.: Ricci curvature of submanifolds in Sasakian space forms. J. Aust. Math. Soc. 72 (2), 247-256 (2002).
• [32] Mustafa, A., Uddin, S., Al-Solamy, F. R.: Chen-Ricci inequality for warped products in Kenmotsu space forms and its applications. RACSAM 113 (4), 3585-3602 (2019).
• [33] Özgür, C., Özgür, N. Y.: Classification of metallic shaped hypersurfaces in real space forms. Turk. J. Math. 39 (5), 784-794 (2015).
• [34] Özgür, C., Özgür, N. Y.: Metallic shaped hypersurfaces in Lorentzian space forms. Rev. Un. Mat. Argentina 58 (2), 215-226 (2017).
• [35] Tripathi, M. M.: Chen-Ricci inequality for submanifolds of contact metric manifolds. J. Adv. Math. Stud. 1 (1-2), 111-134 (2008).
• [36] Vîlcu, G. E.: B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms. Turk. J. Math. 34 (1), 115-128 (2010).
• [37] Yano, K., Kon, M.: Structures on Manifolds. World Scientific, Singapore (1984).
• [38] Yoon, D. W.: Inequality for Ricci curvature of slant submanifolds in cosymplectic space forms. Turk. J. Math. 30 (1), 43-56 (2006)