Differential Geometry Seminar - Ion Mihai

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

1. S. Amari, Differential-Geometrical Methods in Statistics, Springer, Berlin, Germany, 1985.
2. M.E. Aydin, A. Mihai, I. Mihai, Some inequalities on submanifolds in statistical manifolds of constant curvature, Filomat 29 (2015), 465-477.
3. 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.
4. 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.
5. A. Mihai, I. Mihai, Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature, Mathematics 6 (2018), Art. 44.
6. B. Opozda, A sectional curvature for statistical structures, Linear Algebra Appl. 497 (2016), 134-161.