Inequalities for the Casorati Curvature of Statistical Manifolds in Holomorphic Statistical Manifolds of Constant Holomorphic Curvature

Abstract

In this paper, we prove some inequalities in terms of the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of statistical submanifolds in holomorphic statistical manifolds with constant holomorphic sectional curvature. Moreover, we study the equality cases of such inequalities. An example on these submanifolds is presented.

Geometric Inequalities of Warped Product Submanifolds and Their Applications

“…This paper shows the relation between the notion of warped product manifold and homotopy-homology theory. Therefore, we hope that this paper will be of great interest with respect to the topology of Riemannian geometry [28][29][30][31][32][33][34][35] which may find possible applications in physics.…”

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 ].…”

Recent developments in δ-Casorati curvature invariants

“…≤ δ 0 C (r; n − 1) + nC 0 − 2n 2 ∥H 0 ∥ 2 + n 2g (H, H Decu et al also provided in[56] an example which satisfies the equality cases of (18.23) and(18.24) identically.Theorem 18.6 implies the following.…”

“…[56].Let M be an n -dimensional statistical submanifold of a 2m -dimensional holomorphic statistical manifold (M ,∇,g, J ) of constant holomorphic sectional curvature c. Then we have (i) For any real number r such that 0 < r < n(n − 1), 2τ…”