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

Aliya Naaz Siddiqui; Mohammad Hasan Shahid; Jae Won Lee; Aliya Naaz Siddiqui; Mohammad Hasan Shahid; Jae Won Lee

doi:10.3934/math.2020227

AIMS Mathematics

2020, Volume 5, Issue 4: 3495-3509. doi: 10.3934/math.2020227

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Research article

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

Received: 21 January 2020 Accepted: 29 March 2020 Published: 09 April 2020
MSC : 53C05, 53C40, 53A40

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.
- statistical manifolds,
- quasi-constant curvature,
- Ricci curvature,
- Chen-Ricci inequality,
- statistical immersion
Citation: Aliya Naaz Siddiqui, Mohammad Hasan Shahid, Jae Won Lee. On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature[J]. AIMS Mathematics, 2020, 5(4): 3495-3509. doi: 10.3934/math.2020227

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Abstract

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.

References

[1]	B. Y. Chen, Relationship between Ricci curvature and shape operator for submanifolds with arbitrary codimensions, Glasgow Math. J., 41 (1999), 33-41. doi: 10.1017/S0017089599970271
[2]	B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math., 6 (1993), 568-578. doi: 10.1007/BF01236084
[3]	A. Mihai, I. Mihai, Curvature invariants for statistical submanifolds of hessian manifolds of constant hessian curvature, Mathematics, 6 (2018), 44. doi: 10.3390/math6030044
[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. doi: 10.1007/BF00050660
[7]	H. Furuhata, Hypersurfaces in statistical manifolds, Differ. Geom. Appl., 67 (2009), 420-429. doi: 10.1016/j.difgeo.2008.10.019
[8]	B. Opozda, Bochner's technique for statistical structures, Ann. Global Anal. Geom., 48 (2015), 357-395. doi: 10.1007/s10455-015-9475-z
[9]	B. Opozda, A sectional curvature for statistical structures, Linear Algebra Appl., 497 (2016), 134-161. doi: 10.1016/j.laa.2016.02.021
[10]	M. E. Aydin, A. Mihai, I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat, 29 (2015), 465-477. doi: 10.2298/FIL1503465A
[11]	H. Aytimur, C. Ozgur, Inequalities for submanifolds in statistical manifolds of quasi-constant curvature, Ann. Polonici Mathematici, 121 (2018), 197-215. doi: 10.4064/ap171106-27-6
[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. doi: 10.1016/j.jmaa.2019.07.043
[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.

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