Inequalities for the Casorati Curvature of Totally Real Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

Abstract

The purpose of this article is to establish some inequalities concerning the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant) of totally real spacelike submanifolds in statistical manifolds of the type para-Kähler space form. Moreover, this study is focused on the equality cases in these inequalities. Some examples are also provided.

Quasi-Statistical Schouten–van Kampen Connections on the Tangent Bundle

“…Applying Lemma 1, one has T (δ i , δ j , ∂ k ) = 0 for every t ≥ 0 if and only if c 1 d 2 3 = 0. Since c 3 (t) = 0 for every t ≥ 0, from the nondegeneracy condition (6) of the metric G it follows that c 1 (t) = 0 for every t ≥ 0, and hence, the expressions (35) of T (δ i , δ j , ∂ k ) and (32) of T (∂ i , ∂ j , ∂ k ) vanish simultaneously if and only if d 3 = 0, i.e., the metric is of natural diagonal lift type. We compute the other components of the tensor field T with resect to the adapted local frame field {δ i , ∂ j } n i,j=1 by imposing the conditions already obtained, that is c 3 = d 3 = 0 and the locally flatness of the base manifold, and we have that:…”

“…, for every κ 3 ∈ R \ {0}, t > 0, but together with (38), (40), (37), and (43), which would imply d 2 = − c 2 2t , i.e., c 2 + 2td 2 = 0 and d 3 = c 3 , i.e., c 3 + 2td 3 = 0, and hence, the second nondegeneracy condition (6) of the metric G would not be satisfied.…”

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

“…Very recently, Vîlcu studied statistical manifolds endowed with almost product structures and para-Kähler-like statistical submersions [22]. Moreover, Chen et al established Casorati inequalities for totally real spacelike submanifolds in statistical manifolds of type para-Kähler space forms [23].…”

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 ].…”