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

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


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