tailieunhanh - Inequalities for submanifolds of Sasaki-like statistical manifolds

We consider statistical submanifolds in Sasaki-like statistical manifolds. We give some examples of invariant and antiinvariant submanifolds of Sasaki-like statistical manifolds. We prove Chen-like inequality involving scalar curvature and Chen–Ricci inequality for these kinds of submanifolds. | Turk J Math (2018) 42: 3149 – 3163 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article Inequalities for submanifolds of Sasaki-like statistical manifolds Hülya AYTİMUR,, Cihan ÖZGÜR∗, Department of Mathematics, Faculty of Arts and Sciences, Balıkesir University, Balıkesir, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: We consider statistical submanifolds in Sasaki-like statistical manifolds. We give some examples of invariant and antiinvariant submanifolds of Sasaki-like statistical manifolds. We prove Chen-like inequality involving scalar curvature and Chen–Ricci inequality for these kinds of submanifolds. Key words: Sasaki-like statistical manifold, Chen–Ricci inequality, Ricci curvature, scalar curvature 1. Introduction Statistical manifolds have arisen from the study of a statistical distribution. In 1985 Amari [2] introduced a differential geometric approach for a statistical model of discrete probability distribution. Statistical manifolds have many applications in information geometry, which is a branch of mathematics that applies the techniques of differential geometry to the field of probability theory. Some of these applications are statistical inference, linear systems, time series, neural networks, nonlinear systems, linear programming, convex analysis and completely integrable dynamical systems, quantum information geometry, and geometric modeling (for more details see [1]). Let (M, g) be a Riemannian manifold given by a pair of torsion-free affine connections ∇ and ∇∗ . A pair of (∇, g) is called a statistical structure on M if (∇X g) (Y, Z) − (∇Y g) (X, Z) = 0 () holds for X, Y, Z ∈ T M [2]. If a Riemannian manifold (M, g) with its statistical structure satisfies Xg (Y, Z) = g (∇X Y, Z) + g (Y, ∇∗X Z) , then it is called a statistical manifold and denoted by (M, g, ∇, ∇∗ ) (see [2] and [22]). Any torsion-free