On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature

Year 2023, Volume: 16 Issue: 2, 672 - 679, 29.10.2023

Hülya Aytimur

https://doi.org/10.36890/iejg.1237417

Abstract

We mention some properties of statistical submanifolds in statistical
manifolds of quasi-constant curvature. We obtain Chen first inequality and a
Chen inequality for the $\delta (2,2)$-invariant for these manifolds.

Keywords

Statistical manifold of quasi-constant curvature, submanifold, Chen inequality

References

• [1] Amari, S.: Differential-Geometrical Methods in Statistics, Springer-Verlag, (1985).
• [2] Aydın, M. E., Mihai, A., Mihai, I˙ . Some inequalities on submanifolds in statistical manifolds of constant curvature. Filomat. 29 (3), 465-477 (2015).
• [3] Aytimur, H., Özgür, C.: Inequalities for submanifolds in statistical manifolds of quasi-constant curvature. Annales Polonici Mathematici. 121 (3), 197-215 (2018).
• [4] Aytimur, H., Kon, M., Mihai, A., Özgür, C., Takano, K.: Chen inequalities for statistical submanifolds of Kähler-like statistical manifolds. Mathematics. 7 (12), 1202 (2019).
• [5] Aytimur, H., Mihai, A., Özgür, C.: Relations between Extrinsic and Intrinsic Invariants of Statistical Submanifolds in Sasaki-Like Statistical Manifolds. Mathematics. 9 (11), 1285 (2021).
• [6] Chen, B. Y.: Some pinching and classification theorems for minimal submanifolds. Archiv der Mathematic. 60, 568-578 (1993).
• [7] Chen, B.Y.: Mean curvature and shape operator of isometric immersions in real-space-forms. Glasgow Mathematical Journal. 38 (1), 87-97 (1996).
• [8] Chen, B.Y.: Relations between Ricci curvature and shape operator for submanifolds with arbitrary codimensions. Glasgow Mathematical Journal. 41(1), 33-41 (1999).
• [9] Chen, B.Y.: Pseudo-Riemannian Geometry, δ-invariants and Applications. World Scientific Publishing , Hackensack, NJ, (2011).
• [10] 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), 165 (2019).
• [11] Djebbouri, D., Ouakkas, S.: Product of statistical manifolds with doubly warped product. General Mathematics Notes. 31 (2), 16-28 (2015).
• [12] Furuhata, H.: Hypersurfaces in statistical manifolds. Differential Geometry and its Applications. 27 (3), 420-429 (2009).
• [13] Mihai, A.: Modern Topics in Submanifold Theory, Editura Universitatii Bucuresti, Bucharest, (2006).
• [14] Mihai, A., Özgür, C.: Chen inequalities for submanifolds of real space forms with a semi-symmetric metric connection. Taiwanese Journal of Mathematics. 14 (4), 1465-1477 (2010).
• [15] Mihai, A., Özgür, C.: Chen inequalities for submanifolds of complex space forms and Sasakian space forms endowed with semi-symmetric metric connections. Rocky Mountain Journal of Mathematics. 41(5), 1653-1673 (2011).
• [16] Mihai, A., Mihai, I.: Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature. Mathematics. 6 (3), 44 (2018).
• [17] Mihai, A., Mihai, I.: The δ (2, 2) invariant on statistical submanifolds of Hessian manifolds of constant Hessian curvature. Entropy. 22 (2), 164 (2020).
• [18] Mihai, I., Mihai, R. I.: An Algebraic Inequality with Applications to Certain Chen Inequalities. Axioms. 10 (1), 1-7, (2021).
• [19] Opozda, B.: A sectional curvature for statistical structures. Linear Alg. Appl. 497, 134-161 (2016).
• [20] Özgür, C.: B. Y. Chen inequalities for submanifolds of a Riemanian manifold of quasi-constant curvature. Turkish Journal of Mathematics. 35, 501-509 (2011).
• [21] Todjihounde, L.: Dualistic structure on warped product manifolds. Differential Geometry-Dynamical Systems. 8, 278-284 (2006).
• [22] Vos, P. W.: Fundamental equations for statistical submanifolds with applications to the Bartlett correction, Annals of the Institute of Statististical Mathematics. 41 (3), 429-450 (1989).