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Chen invariants and statistical submanifolds
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We define a kind of sectional curvature and $\delta$-invariants for statistical manifolds. For statistical submanifolds the sum of the squared mean curvature and the squared dual mean curvature is bounded below by using the $\delta$-invariant. This inequality can be considered as a generalization of the so-called Chen inequality for Riemannian submanifolds.
Abstract
We define a kind of sectional curvature and $\delta$-invariants for statistical manifolds. For statistical submanifolds the sum of the squared mean curvature and the squared dual mean curvature is bounded below by using the $\delta$-invariant. This inequality can be considered as a generalization of the so-called Chen inequality for Riemannian submanifolds.
참고문헌 (10)
- Muhittin Aydin, "Some inequalities on submanifolds in statistical manifolds of constant curvature", Filomat, 29(3), pp.465~477 (2015), National Library of Serbia[DOI]
- B. -Y. Chen, "A general optimal inequality for arbitrary Riemannian submanifolds", JI-PAM. J. Inequal. Pure Appl. Math, 6(3), pp.10~ (2005)
- Bang-Yen Chen, "A Chen First Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature", Results in Mathematics, 74(4), pp.11~ (2019), Springer Science and Business Media LLC[DOI]
- H. Furuhata, "Geometry of Cauchy-Riemann submanifolds", pp.179~215 (2016), Springer
- Hitoshi Furuhata, "Sasakian statistical manifolds", Journal of Geometry and Physics, 117, pp.179~186 (2017), Elsevier BV[DOI]
- Chul Woo Lee, "A pinching theorem for statistical manifolds with Casorati curvatures", The Journal of Nonlinear Sciences and Applications, 10(09), pp.4908~4914 (2017), International Scientific Research Publications MY SDN. BHD.[DOI]
- Adela Mihai, "The δ(2,2)-Invariant on Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature", Entropy, 22(2), pp.164~ (2020), MDPI AG[DOI]
- Barbara Opozda, "A sectional curvature for statistical structures", Linear Algebra and its Applications, 497, pp.134~161 (2016), Elsevier BV[DOI]
- Naoto Satoh, "Statistical submanifolds from a viewpoint of the Euler inequality", Information Geometry, 4(1), pp.189~213 (2020), Springer Science and Business Media LLC[DOI]
- H. Shima, "The geometry of Hessian structures" (2007), World Scienti c Publishing Co. Pte. Ltd
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