scholarly journals Inequalities on Sasakian Statistical Manifolds in Terms of Casorati Curvatures

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 259 ◽  
Author(s):  
Chul Lee ◽  
Jae Lee
Keyword(s):  
Scalar Curvature ◽  
Riemannian Metric ◽  
Statistical Manifold ◽  
Statistical Structure ◽  
Slant Submanifolds ◽  
Statistical Manifolds ◽  
Conjugate Connections ◽  
Levi Civita Connection ◽  
Normalized Scalar Curvature ◽  
Totally Real

A statistical structure is considered as a generalization of a pair of a Riemannian metric and its Levi-Civita connection. With a pair of conjugate connections ∇ and ∇ * in the Sasakian statistical structure, we provide the normalized scalar curvature which is bounded above from Casorati curvatures on C-totally real (Legendrian and slant) submanifolds of a Sasakian statistical manifold of constant φ -sectional curvature. In addition, we give examples to show that the total space is a sphere.

scholarly journals Statistical Solitons and Inequalities for Statistical Warped Product Submanifolds

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 797 ◽  
Author(s):  
Aliya Siddiqui ◽  
Bang-Yen Chen ◽  
Oğuzhan Bahadır
Keyword(s):  
Differential Geometry ◽  
Scalar Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Mathematical Physics ◽  
Warping Function ◽  
Warped Products ◽  
Statistical Manifold ◽  
Statistical Manifolds ◽  
Levi Civita Connection

Warped products play crucial roles in differential geometry, as well as in mathematical physics, especially in general relativity. In this article, first we define and study statistical solitons on Ricci-symmetric statistical warped products R × f N 2 and N 1 × f R . Second, we study statistical warped products as submanifolds of statistical manifolds. For statistical warped products statistically immersed in a statistical manifold of constant curvature, we prove Chen’s inequality involving scalar curvature, the squared mean curvature, and the Laplacian of warping function (with respect to the Levi–Civita connection). At the end, we establish a relationship between the scalar curvature and the Casorati curvatures in terms of the Laplacian of the warping function for statistical warped product submanifolds in the same ambient space.

Some inequalities of slant submanifolds in generalized complex space forms

2005 ◽  
Vol 36 (3) ◽  
pp. 223-229 ◽  
Author(s):  
Aimin Song ◽  
Ximin Liu
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Complex Space ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Generalized Complex ◽  
Complex Space Forms ◽  
Normalized Scalar Curvature

In this paper, we obtain an inequality about Ricci curvature and squared mean curvature of slant submanifolds in generalized complex space forms. We also obtain an inequality about the squared mean curvature and the normalized scalar curvature of slant submanifolds in generalized coplex space forms.

Lower Bounds on Statistical Submersions with vertical Casorati curvatures

2021 ◽  
Author(s):  
Aliya Naaz Siddiqui ◽  
Mohd Danish Siddiqi ◽  
Ali H. Alkhaldi ◽  
Akram Ali
Keyword(s):  
Scalar Curvature ◽  
Lower Bounds ◽  
Statistical Manifolds ◽  
Normalized Scalar Curvature ◽  
Statistical Submersion

In this paper, we obtain lower bounds for the normalized scalar curvature on statistical submersion with the normalized [Formula: see text]-vertical Casorati curvatures. Also, we discuss the conditions for which the equality cases hold. Beside this, we determine the statistical solitons on statistical submersion from statistical manifolds and illustrate an example of statistical submersions from statistical manifolds.

scholarly journals The Faraday 2-form in Einstein-Weyl geometry

2001 ◽  
Vol 89 (1) ◽  
pp. 97 ◽  
Author(s):  
David M. J. Calderbank
Keyword(s):  
Scalar Curvature ◽  
Higher Dimensions ◽  
Riemannian Metric ◽  
Constant Sign ◽  
Weyl Curvature ◽  
Weyl Geometry ◽  
Four Dimensions ◽  
Bach Tensor ◽  
Levi Civita Connection ◽  
Torsion Free

On a conformal manifold, a compatible torsion free connection $D$ need not be the Levi-Civita connection of a compatible Riemannian metric. The local obstruction is a real $2$-form $F^D$, the Faraday curvature. It is shown that, except in four dimensions, $F^D$ necessarily vanishes if it is divergence free. In four dimensions another differential operator may be applied to $F^D$ to show that an Einstein-Weyl $4$-manifold with selfdual Weyl curvature also has selfdual Faraday curvature and so is either Einstein or locally hypercomplex. More generally, the Bach tensor and the scalar curvature are shown to control the selfduality of $F^D$. Finally, the constancy of the sign of the scalar curvature on compact Einstein-Weyl $4$-manifolds [24] is generalised to higher dimensions. The scalar curvature need not have constant sign in dimensions two and three.

scholarly journals Totally real submanifolds of (LCS)n-manifolds

2018 ◽  
Vol 33 (2) ◽  
pp. 141
Author(s):  
Shyamal Kumar Hui ◽  
Tanumoy Pal
Keyword(s):  
Scalar Curvature ◽  
Real Submanifolds ◽  
Civita Connection ◽  
Metric Connection ◽  
Levi Civita Connection ◽  
Totally Real

The present paper deals with the study of totally real submanifolds and C-totally real submanifolds of (LCS)n-manifolds withrespect to Levi-Civita connection as well as quarter symmetric metric connection. It is proved that scalar curvature of C-totally real submanifolds of (LCS)n-manifold with respect to both the said connections are same.

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 Optimal casorati inequalities on bi-slant submanifolds of generalized sasakian space forms

2018 ◽  
Vol 49 (3) ◽  
pp. 235-255 ◽  
Author(s):  
Aliya Naaz Siddiqui
Keyword(s):  
Scalar Curvature ◽  
Optimization Method ◽  
Space Forms ◽  
Slant Submanifolds ◽  
Sasakian Space Forms ◽  
Normalized Scalar Curvature

In this paper, we use T Oprea's optimization method to establish some optimal Casorati inequalities, which involve the normalized scalar curvature for bi-slant submanifolds of generalized Sasakian space forms. In the continuation, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical.

scholarly journals Bounds for generalized normalized δ-Casorati curvatures for Bi-slant submanifolds in T-space forms

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 329-340 ◽  
Author(s):  
Mohd. Aquib
Keyword(s):  
Scalar Curvature ◽  
Space Forms ◽  
Equality Case ◽  
Slant Submanifolds ◽  
Invariant Submanifolds ◽  
Normalized Scalar Curvature ◽  
Cr Submanifolds

In this paper, we prove the inequality between the generalized normalized ?-Casorati curvatures and the normalized scalar curvature for the bi-slant submanifolds in T-space forms and consider the equality case of the inequality. We also develop same results for semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds, slant submanifolds, invariant and anti-invariant submanifolds in T-space forms.

scholarly journals Bounds for Statistical Curvatures of Submanifolds in Kenmotsu-like Statistical Manifolds

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 176
Author(s):  
Aliya Naaz Siddiqui ◽  
Mohd Danish Siddiqi ◽  
Ali Hussain Alkhaldi
Keyword(s):  
Scalar Curvature ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
The Second Fundamental Form ◽  
Statistical Manifolds ◽  
Equality Condition ◽  
Normalized Scalar Curvature ◽  
Legendrian Submanifolds

In this article, we obtain certain bounds for statistical curvatures of submanifolds with any codimension of Kenmotsu-like statistical manifolds. In this context, we construct a class of optimum inequalities for submanifolds in Kenmotsu-like statistical manifolds containing the normalized scalar curvature and the generalized normalized Casorati curvatures. We also define the second fundamental form of those submanifolds that satisfy the equality condition. On Legendrian submanifolds of Kenmotsu-like statistical manifolds, we discuss a conjecture for Wintgen inequality. At the end, some immediate geometric consequences are stated.

scholarly journals Optimizations on Statistical Hypersurfaces with Casorati Curvatures

2021 ◽  
Vol 45 (03) ◽  
pp. 449-463
Author(s):  
ALIYA NAAZ SIDDIQUI ◽  
MOHAMMAD HASAN SHAHID
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Real Hypersurface ◽  
Holomorphic Sectional Curvature ◽  
Statistical Manifold ◽  
Equality Case ◽  
Normalized Scalar Curvature

In the present paper, we study Casorati curvatures for statistical hypersurfaces. We show that the normalized scalar curvature for any real hypersurface (i.e., statistical hypersurface) of a holomorphic statistical manifold of constant holomorphic sectional curvature k is bounded above by the generalized normalized δ−Casorati curvatures and also consider the equality case of the inequality. Some immediate applications are discussed.

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