Some Inequalities for Statistical Submanifolds in Metallic-like Statistical Manifolds

Year 2021, Volume: 13 Issue: 2, 348 - 358, 31.12.2021

Oğuzhan Bahadır

Abstract

In this study, we introduce metallic-like statistical manifolds and provide some examples, which are a generalized version of metallic manifolds. We also obtain the first Chen inequality and a Chen inequality for the $\delta(2,2)$ invariant for statistical submanifolds in metallic-like statistical manifolds.

Keywords

Chen invariants , Statistical manifolds , Metallic manifolds.

References

• [1] Amari, S., Differential-geometrical methods in statistics, Springer Science & Business Media, 28(2012).
• [2] Alkhaldi, A.H., Aquib, M., Siddiqui, A.N., Shahid, M.H., Pinching theorems for statistical submanifolds in Sasaki-like statistical space forms, Entropy, 20(9)(2018), 690, https://doi.org/10.3390/e20090690.
• [3] Aquib, M., On Some inequalities for statistical submanifolds of quaternion Kaehler-Like statistical space forms, International Journal of Geometric Methods in Modern Physics, 16(8)(2019), 1950129.
• [4] Aquib, M., Boyom, M.N., Alkhaldi, A.H., Shahid, M.H., B.Y. Chen inequalities for statistical submanifolds in Sasakian statistical manifolds, In: Nielsen F., Barbaresco F. (eds) Geometric Science of Information. GSI 2019. Lecture Notes in Computer Science Springer, Cham., 11712(2019).
• [5] Aquib, M., Shahid, M.H., Generalized normalized $\delta$-Casorati curvature for statistical submanifolds in quaternion Kaehler-like statistical space forms, J. Geom., 109(2018), 1–13.
• [6] Aydin, M.E., Mihai, A., Mihai, I., Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat, 29(3)(2015), 465–477.
• [7] Aytimur, H., Kon, M., Mihai, A., Özgür, C., Takano, K., Chen inequalities for statistical submanifolds of Kahler-like statistical manifolds, Mathematics, 7(12)(2019), 1202.
• [8] Blaga, A.M., Hretcanu, C.E., Invariant, anti-invariant and slant submanifolds of a metallic Riemannian manifold, Novi Sad J. Math., 48(2)(2018), 55–80.
• [9] Choudhary, M.A., Blaga, A.M., Inequalities for generalized normalized $\delta$-Casorati curvatures of slant submanifolds in metallic Riemannian space forms , J. Geom., 111(2020), 39.
• [10] Chen, B.Y., Mean curvature and shape operator of isometric immersions in real-space-forms, Glasgow Mathematical Journal, 38(1)(1996), 87–97.
• [11] Chen, B.Y., Relations between ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Mathematical Journal, 41(1)(1999), 33–41.
• [12] Chen, B.Y., Mihai, A., Mihai, I., A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature, Results in Mathematics, 74(4)(2019), 165.
• [13] Goldberg, S.I., Yano, K., Polynomial structures on manifolds, Kodai Mathematical Seminar Reports, 2(1970), 199–218.
• [14] Hretcanu, C.E., Blaga, A.M., Submanifolds in metallic Riemannian manifolds, Differential Geometry–Dynamical Systems, 20(2018).
• [15] Hretcanu, C.E., Crasmareanu, M., Applications of the Golden ratio on Riemannian manifolds, Turkish Journal of Mathematics, 33(2)(2009), 179–191.
• [16] Hretcanu, C.E., Crasmareanu, M., Metallic structures on Riemannian manifolds, Rev. Un. Mat. Argentina, 54(2)(2013), 15–27.
• [17] lounesto, P., Clifford Algebras and Spinors, Cambridge University Press, United Kingdom, 2001.
• [18] Macsim, G., Mihai, A., Mihai, I., $\delta$(2, 2)-invariant for lagrangian submanifolds in quaternionic space forms, Mathematics, 8(4)(2020), 480.
• [19] Manea, A., Some remarks on metallic Riemannian structures, An. S¸ tiint. Univ. Al. I. Cuza Ias¸i. Mat., 65(1)(2019), 37–46.
• [20] Mihai, A., Özgür, C., Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections, The Rocky Mountain Journal of Mathematics, (2011), 1653–1673.
• [21] Mihai, A., Mihai, I., Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature, Mathematics, 6(3)(2018).
• [22] Mihai, A., Mihai, I., The $\delta$ (2, 2)-invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature, Entropy, 22(2)(2020), 164.
• [23] Mustafa, A., Uddin, S., Al-Solamy, F.R., Chen–ricci inequality for warped products in kenmotsu space forms and its applications, Revista de la Real Academia de Ciencias Exactas F´ısicas y Naturales, 113(4)(2019), 3585–3602.
• [24] Nomizu, K., Katsumi, N., Sasaki, T., Ane Dierential Geometry: Geometry of Affine Immersions, Cambridge university press, 1994.
• [25] Opozda, B., A sectional curvature for statistical structures, Linear Algebra and its Applications, 38(497)(2016), 134–161.
• [26] Oprea, T., Chen’s inequality in the Lagrangian case, Colloquium Mathematicum, 108(2007), 163–169.
• [27] Ozkan, M., Yilmaz, F., Metallic structures on differentiable manifolds, Journal of Science and Arts, 44(3)(2018), 645–660
• [28] Siddiqui, A.N., Al-Solamy, F.R., Shahid, M.H., Mihai, I., On CR-statistical submanifolds of holomorphic statistical manifolds, arXiv e-prints, (2020).
• [29] Spinadel, V., On characterization of the onset to chaos, Chaos, Solitons and Fractals, 8(10)(1997), 1631–1643.
• [30] Spinadel, V., The metallic means family and multifractal spectra, Nonlinear Analysis, 36 (1999), 721–745.
• [31] Spinadel, V., The metallic means family and forbidden symmetries, Int Math J., 2(3)(2002), 279–88.
• [32] Takano, K., Statistical manifolds with almost complex structures and its statistical submersions, Tensor. New series, 65(2)(2004), 128–142.
• [33] Yano, K., Kon, M., Structures on Manifolds, World Scientific, 1984.
• [34] Vilcu, A.D., Vilcu, G.E., Statistical manifolds with almost quaternionic structures and quaternionic Kahler-like statistical submersions, Entropy, 17(3)(2015), 6213–6228.