Generalized normalized $$\varvec{\delta }$$ δ -Casorati curvature for statistical submanifolds in quaternion Kaehler-like statistical space forms

2018 ◽  
Vol 109 (1) ◽  
Author(s):  
Mohd. Aquib ◽  
Mohammad Hasan Shahid
Keyword(s):  
Space Forms ◽  
Casorati Curvature ◽  
Statistical Space

Certain inequalities for the Casorati curvatures of submanifolds of generalized (k,μ)-space-forms

2018 ◽  
Vol 13 (02) ◽  
pp. 2050040
Author(s):  
Shyamal Kumar Hui ◽  
Pradip Mandal ◽  
Ali H. Alkhaldi ◽  
Tanumoy Pal
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Space Forms ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Casorati Curvature ◽  
Optimal Inequalities

The paper deals with the study of Casorati curvature of submanifolds of generalized [Formula: see text]-space-form with respect to Levi-Civita connection as well as semisymmetric metric connection and derived two optimal inequalities between scalar curvature and Casorati curvature of such space forms. The equality cases are also considered.

scholarly journals Pinching Theorems for Statistical Submanifolds in Sasaki-Like Statistical Space Forms

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 690 ◽  
Author(s):  
Ali Alkhaldi ◽  
Mohd. Aquib ◽  
Aliya Siddiqui ◽  
Mohammad Shahid
Keyword(s):  
Constant Curvature ◽  
Necessary And Sufficient Condition ◽  
Einstein Field Equations ◽  
Statistical Manifold ◽  
Field Equations ◽  
Space Forms ◽  
Equality Case ◽  
Statistical Manifolds ◽  
Necessary And Sufficient ◽  
Statistical Space

In this paper, we obtain the upper bounds for the normalized δ -Casorati curvatures and generalized normalized δ -Casorati curvatures for statistical submanifolds in Sasaki-like statistical manifolds with constant curvature. Further, we discuss the equality case of the inequalities. Moreover, we give the necessary and sufficient condition for a Sasaki-like statistical manifold to be η -Einstein. Finally, we provide the condition under which the metric of Sasaki-like statistical manifolds with constant curvature is a solution of vacuum Einstein field equations.

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

Entropy ◽  
2021 ◽  
Vol 23 (11) ◽  
pp. 1399
Author(s):  
Bang-Yen Chen ◽  
Simona Decu ◽  
Gabriel-Eduard Vîlcu
Keyword(s):  
Scalar Curvature ◽  
Space Form ◽  
Space Forms ◽  
Statistical Manifolds ◽  
Spacelike Submanifolds ◽  
Totally Real ◽  
Casorati Curvature

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.

Some inequalities for statistical submanifolds of quaternion Kaehler-like statistical space forms

2019 ◽  
Vol 16 (08) ◽  
pp. 1950129 ◽  
Author(s):  
Mohd. Aquib
Keyword(s):  
Space Forms ◽  
Statistical Manifolds ◽  
Statistical Submanifold ◽  
Totally Real ◽  
Statistical Version ◽  
Chen Inequality ◽  
Statistical Space

Motivated by one of the problems proposed by [Vilcu and Vilcu, Statistical manifolds with almost quaternionic structures and quaternionic Kaehler-like statistical submersions, Entropy 17 (2015) 6213–6228] in this paper, we study the statistical submanifolds of quaternion Kaehler-like statistical space forms and provide an answer to the problem. Further, we derive the statistical version of Chen inequality for totally real statistical submanifold in such ambient.

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