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On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature

Year 2023, , 672 - 679, 29.10.2023
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.

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).
Year 2023, , 672 - 679, 29.10.2023
https://doi.org/10.36890/iejg.1237417

Abstract

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).
There are 22 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Hülya Aytimur 0000-0003-4420-9861

Early Pub Date October 19, 2023
Publication Date October 29, 2023
Acceptance Date May 24, 2023
Published in Issue Year 2023

Cite

APA Aytimur, H. (2023). On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature. International Electronic Journal of Geometry, 16(2), 672-679. https://doi.org/10.36890/iejg.1237417
AMA Aytimur H. On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature. Int. Electron. J. Geom. October 2023;16(2):672-679. doi:10.36890/iejg.1237417
Chicago Aytimur, Hülya. “On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature”. International Electronic Journal of Geometry 16, no. 2 (October 2023): 672-79. https://doi.org/10.36890/iejg.1237417.
EndNote Aytimur H (October 1, 2023) On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature. International Electronic Journal of Geometry 16 2 672–679.
IEEE H. Aytimur, “On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature”, Int. Electron. J. Geom., vol. 16, no. 2, pp. 672–679, 2023, doi: 10.36890/iejg.1237417.
ISNAD Aytimur, Hülya. “On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature”. International Electronic Journal of Geometry 16/2 (October 2023), 672-679. https://doi.org/10.36890/iejg.1237417.
JAMA Aytimur H. On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature. Int. Electron. J. Geom. 2023;16:672–679.
MLA Aytimur, Hülya. “On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature”. International Electronic Journal of Geometry, vol. 16, no. 2, 2023, pp. 672-9, doi:10.36890/iejg.1237417.
Vancouver Aytimur H. On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature. Int. Electron. J. Geom. 2023;16(2):672-9.