Lecture Details
Name: Ion Mihai (University of Bucharest, Romania)Speaking: Thu 9th May 15:00 - 15:25Title: Hessian manifolds and their submanifoldsAbstract: Hessian manifolds are statistical manifolds of constant curvature 0. Statistical manifolds were introduced by S. Amari [1].
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 [5].
Recently, in co-operation with A. Mihai [4], we established a Euler inequality and a Chen-Ricci inequality for submanifolds in Hessian manifolds of constant Hessian curvature.
We shall continue the study of Chen-like invariants on such submanifolds.\bigskip
{\bf References}\smallskip
[1] S. Amari, {\it Differential-Geometrical Methods in Statistics}, Springer, Berlin, Germany, 1985.
[2] M.E. Aydin, A. Mihai, I. Mihai, {\it Some inequalities on submanifolds in statistical manifolds of constant curvature}, Filomat {\bf 29} (2015), 465-477.
[3] M.E. Aydin, A. Mihai, I. Mihai, {\it Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature}, Bull. Math. Sci. {\bf 7} (2017), 155-166.
[4] A. Mihai, I. Mihai, {\it Curvature invariants for statistical submanifolds of Hessian manifolds of constant Hessian curvature}, Mathematics {\bf 6} (2018), Art. 44.
[5] B. Opozda, {\it A sectional curvature for statistical structures}, Linear Algebra Appl. {\bf 497} (2016), 134-161.