Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

Abstract

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality cases of these inequalities. As illustrations of the applications of the above inequalities, we consider a few examples.

General Chen Inequalities for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

“…Chen et al [10]. After that, particular cases of Chen inequalities in statistical settings were obtained (see [11][12][13][14][15][16][17][18]).…”

Casorati Inequalities for Statistical Submanifolds in Kenmotsu Statistical Manifolds of Constant ϕ-Sectional Curvature with Semi-Symmetric Metric Connection

“…The study of simple relationships between the main intrinsic and extrinsic invariants of submanifolds is a fundamental problem in submanifold theory [ 1 ]. Recent research shows a growing trend in approaching this fascinating problem through an approach that proves some types of geometric inequalities (see, e.g., [ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ]).…”

“…The topic of -Casorati curvatures will appeal to more geometers focused on finding new solutions of the above problem. In this respect, some recent developments are devoted to inequalities on various submanifolds of a statistical manifold , notion defined by Amari [ 18 ] in 1985 in the realm of information geometry [ 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 ]. In this setting, the Fisher information metric is one of the most important metrics that can be considered on statistical models [ 19 ].…”

Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

“…Afterward, many authors obtained Chen's inequalities for different submanifolds in various ambient spaces, such as the Kenmotsu space form [4], the Sasakian-space-form [5], the Cosympletic space form [6], the Riemannian manifold of quasi-constant curvature [7], generalized space forms [8,9], Statistical manifolds [10][11][12], quaternionic space forms [13] and the GRW spacetime [14].…”