Title: Curvature Invariants on Statistical Manifolds and thier Submanifolds
Speaker: Ion Mihai (University of Bucharest, Romania)
Abstract: Statistical manifolds were introduced by S. Amari [1]. In particular, Hessian manifolds are statistical manifolds of constant curvature 0. The geometry of statistical manifolds and their submanifolds is a modern topic of research in pure and applied mathematics. M.E. Aydin, A. Mihai and the present author [2] obtained geometric inequalities for the scalar curvature and Ricci curvature associated to the dual connections for submanifolds in statistical manifolds of constant curvature. In [3], the same authors proved a generalized Wintgen inequality for such submanifolds, with respect to a sectional curvature introduced by B. Opozda [6]. Recently, in co-operation with A. Mihai [5], we established a Euler inequality and a Chen-Ricci inequality for submanifolds in Hessian manifolds of constant Hessian curvature. Recently, we proved Chen first inequality on such submanifolds (see [4]). The present talk is a survey on basic notions and recent results in this topic.
References
- S. Amari, Differential-Geometrical Methods in Statistics, Springer, Berlin, Germany, 1985.
- M.E. Aydin, A. Mihai, I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat 29 (2015), 465-477.
- M.E. Aydin, A. Mihai, I. Mihai, Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature, Bull. Math. Sci. 7 (2017), 155-166.
- B.Y. Chen, A. Mihai, I. Mihai, A Chen first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature, Results Math., to appear.
- A. Mihai, I. Mihai, Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature, Mathematics 6 (2018), Art. 44.
- B. Opozda, A sectional curvature for statistical structures, Linear Algebra Appl. 497 (2016), 134-161.
