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 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 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 Lower bounds on Ricci flow invariant curvatures and geometric applications

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Thomas Richard
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Nonnegative Ricci Curvature ◽  
Invariant Cones

AbstractWe consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “nonnegative Ricci curvature” and “nonnegative curvature operator”. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies

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.

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.

On the normal scalar curvature conjecture in Kenmotsu statistical manifolds

2019 ◽  
Vol 142 ◽  
pp. 37-46 ◽  
Author(s):  
Pooja Bansal ◽  
Siraj Uddin ◽  
Mohammad Hasan Shahid
Keyword(s):  
Scalar Curvature ◽  
Statistical Manifolds

A lower bound of normalized scalar curvature for the submanifolds of locally conformal Kaehler space form using Casorati curvatures

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4925-4932 ◽  
Author(s):  
Mehraj Lone
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Space Form ◽  
Equality Case ◽  
Locally Conformal ◽  
Normalized Scalar Curvature

In the present paper, we prove the inequality between the normalized scalar curvature and the generalized normalized ?-Casorati curvatures for the submanifolds of locally conformal Kaehler space form and also consider the equality case of the inequality.

scholarly journals On the lower bounds of the $L^2$-norm of the Hermitian scalar curvature

2020 ◽  
Vol 18 (2) ◽  
pp. 537-558
Author(s):  
Julien Keller ◽  
Mehdi Lejmi
Keyword(s):  
Scalar Curvature ◽  
Lower Bounds
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