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.

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.

scholarly journals On the geometry of lightlike submanifolds of indefinite statistical manifolds

2020 ◽  
Vol 17 (07) ◽  
pp. 2050099
Author(s):  
Varun Jain ◽  
Amrinder Pal Singh ◽  
Rakesh Kumar
Keyword(s):  
Sectional Curvature ◽  
Classical Theory ◽  
Ricci Tensor ◽  
Statistical Manifolds ◽  
Lightlike Submanifolds ◽  
Statistical Submanifold ◽  
Lightlike Submanifold

We study lightlike submanifolds of indefinite statistical manifolds. Contrary to the classical theory of submanifolds of statistical manifolds, lightlike submanifolds of indefinite statistical manifolds need not to be statistical submanifold. Therefore, we obtain some conditions for a lightlike submanifold of indefinite statistical manifolds to be a lightlike statistical submanifold. We derive the expression of statistical sectional curvature and finally obtain some conditions for the induced statistical Ricci tensor on a lightlike submanifold of indefinite statistical manifolds to be symmetric.

scholarly journals On totally real statistical submanifolds

Filomat ◽  
2018 ◽  
Vol 32 (13) ◽  
pp. 4473-4483
Author(s):  
Aliya Siddiqui ◽  
Mohammad Shahid
Keyword(s):  
Sectional Curvature ◽  
Holomorphic Sectional Curvature ◽  
Fundamental Properties ◽  
Statistical Manifolds ◽  
Totally Real

In the present paper, first we prove some results by using fundamental properties of totally real statistical submanifolds immersed into holomorphic statistical manifolds. Further, we obtain the generalizedWintgen inequality for Lagrangian statistical submanifolds of holomorphic statistical manifolds with constant holomorphic sectional curvature c. The paper finishes with some geometric consequences of obtained results.

scholarly journals A class of C-totally real submanifolds of Sasakian space forms

1998 ◽  
Vol 65 (1) ◽  
pp. 120-128
Author(s):  
Filip Defever ◽  
Ion Mihai ◽  
Leopold Verstraelen
Keyword(s):  
Riemannian Manifold ◽  
Mean Curvature ◽  
Space Form ◽  
Sasakian Space Form ◽  
Real Submanifolds ◽  
Space Forms ◽  
Sasakian Space Forms ◽  
Totally Real ◽  
Real Complex

AbstractRecently, Chen defined an invariant δM of a Riemannian manifold M. Sharp inequalities for this Riemannian invariant were obtained for submanifolds in real, complex and Sasakian space forms, in terms of their mean curvature. In the present paper, we investigate certain C-totally real submanifolds of a Sasakian space form M2m+1(C)satisfying Chen's equality.

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