On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature
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...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2020-05-01
|
| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/math.2020227/fulltext.html |
| _version_ | 1828452778555473920 |
|---|---|
| author | Aliya Naaz Siddiqui Mohammad Hasan Shahid Jae Won Lee |
| author_facet | Aliya Naaz Siddiqui Mohammad Hasan Shahid Jae Won Lee |
| author_sort | Aliya Naaz Siddiqui |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-10T23:58:07Z |
| format | Article |
| id | doaj.art-2866e74ebc0e4180a9ce20ea9f121a31 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-10T23:58:07Z |
| publishDate | 2020-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-2866e74ebc0e4180a9ce20ea9f121a312022-12-22T01:28:32ZengAIMS PressAIMS Mathematics2473-69882020-05-01543495350910.3934/math.2020227On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvatureAliya Naaz Siddiqui0Mohammad Hasan Shahid1Jae Won Lee21 Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India1 Department of Mathematics, Jamia Millia Islamia, New Delhi-110025, India2 Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of KoreaIn 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.https://www.aimspress.com/article/10.3934/math.2020227/fulltext.htmlstatistical manifoldsquasi-constant curvaturericci curvaturechen-ricci inequalitystatistical immersion |
| spellingShingle | Aliya Naaz Siddiqui Mohammad Hasan Shahid Jae Won Lee On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature AIMS Mathematics statistical manifolds quasi-constant curvature ricci curvature chen-ricci inequality statistical immersion |
| title | On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature |
| title_full | On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature |
| title_fullStr | On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature |
| title_full_unstemmed | On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature |
| title_short | On Ricci curvature of submanifolds in statistical manifolds of constant (quasi-constant) curvature |
| title_sort | on ricci curvature of submanifolds in statistical manifolds of constant quasi constant curvature |
| topic | statistical manifolds quasi-constant curvature ricci curvature chen-ricci inequality statistical immersion |
| url | https://www.aimspress.com/article/10.3934/math.2020227/fulltext.html |
| work_keys_str_mv | AT aliyanaazsiddiqui onriccicurvatureofsubmanifoldsinstatisticalmanifoldsofconstantquasiconstantcurvature AT mohammadhasanshahid onriccicurvatureofsubmanifoldsinstatisticalmanifoldsofconstantquasiconstantcurvature AT jaewonlee onriccicurvatureofsubmanifoldsinstatisticalmanifoldsofconstantquasiconstantcurvature |