AIMS Mathematics, 2020, 5(4): 3495-3509. doi: 10.3934/math.2020227

Research article

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature

1 Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India
2 Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea

In 1999, B. Y. Chen established a sharp inequality between the Ricci curvature and the squared mean curvature for an arbitrary Riemannian submanifold of a real space form. This inequality was extended in 2015 by M. E. Aydin et al. to the case of statistical submanifolds in a statistical manifold of constant curvature, obtaining a lower bound for the Ricci curvature of the dual connections. Also, the similar inequality for submanifolds in statistical manifolds of quasi-constant curvature studied by H. Aytimur and C. Ozgur in their recent article. In the present paper, we give a different proof of the same inequality but working with the statistical curvature tensor field, instead of the curvature tensor fields with respect to the dual connections. A geometric inequality can be treated as an optimization problem. The new proof is based on a simple technique, known as Oprea’s optimization method on submanifolds, namely analyzing a suitable constrained extremum problem. We also provide some examples. This paper finishes with some conclusions and remarks.
  Article Metrics


1. B. Y. Chen, Relationship between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J., 41 (1999), 33-41.    

2. B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math., 6 (1993), 568-578.

3. A. Mihai, I. Mihai, Curvature invariants for statistical submanifolds of hessian manifolds of constant hessian curvature, Mathematics, 6 (2018), 44.

4. A. N. Siddiqui, Y. J. Suh, O. Bahadir, Extremities for statistical submanifolds in Kenmotsu statistical manifolds, 2019.

5. S. Amari, Differential-Geometrical methods in statistics, lecture notes in statistics, Springer: New York, NY, USA, 1985.

6. P. W. Vos, Fundamental equations for statistical submanifolds with applications to the Bartlett correction, Ann. Inst. Stat. Math., 41 (1989), 429-450.    

7. H. Furuhata, Hypersurfaces in statistical manifolds, Differ. Geom. Appl., 67 (2009), 420-429.

8. B. Opozda, Bochner's technique for statistical structures, Ann. Global Anal. Geom., 48 (2015), 357-395.    

9. B. Opozda, A sectional curvature for statistical structures, Linear Algebra Appl., 497 (2016), 134-161.    

10. M. E. Aydin, A. Mihai, I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat, 29 (2015), 465-477.    

11. H. Aytimur, C. Ozgur, Inequalities for submanifolds in statistical manifolds of quasi-constant curvature, Ann. Polonici Mathematici, 121 (2018), 197-215.    

12. T. Oprea, On a geometric inequality, arXiv:math/0511088v1[math.DG], 2005.

13. T. Oprea, Optimizations on riemannian submanifolds, An. Univ. Bucureşti Mat., 54 (2005), 127-136.

14. L. Peng, Z. Zhang, Statistical Einstein manifolds of exponential families with group-invariant potential functions, J. Math. Anal. and App., 479 (2019), 2104-2118.    

15. A. Rylov, Constant curvature connections on statistical models, In: Ay N., Gibilisco P., Matúš F. Information geometry and its applications. IGAIA IV 2016. Eds. Springer Proceedings in Mathematics Statistics, Springer, Cham, 252 (2018), 349-361.

16. K. Arwini, C. T. J. Dodson, Information geometry: Near randomness and near independence, lecture notes in mathematics, Springer-Verlag Berlin Heidelberg, 1953 (2008), 260.

© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved