Chen Inequality Research Articles

Chen's Inequality for CR-warped Products in Locally Metallic Riemannian Manifolds

In this paper, we study CR warped product submanifolds in locally metallic Riemannian manifolds. We provide several non-trivial examples of such submanifolds. We establish a sharp inequality known as Chen's inequality for the squared norm of the second fundamental form. We also discuss the equality case of Chen's inequality.

Riemannian invariants for warped product submanifolds in Q ε m × R {{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}} and their applications

Abstract This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product Q ε m × R {{\mathbb{Q}}}_{\varepsilon }^{m}\times {\mathbb{R}} . In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and δ \delta -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.

Chen-Type Inequality for Generic Submanifolds of Quaternionic Space Form and Its Application

In 1993, the theory of Chen invariants started when Chen wrote basic inequalities for submanifolds in space forms. This inequality is called Chen’s first inequality. Afterward, many geometers studied many papers dealing with this new inequality. The present paper aims to establish a Chen inequality for quaternionic generic submanifolds in a quaternionic space form and obtain this inequality for real hypersurfaces.

An optimized Chen first inequality for semi-slant submanifolds in Lorentz Kenmotsu space forms

In this paper, we present necessary and sufficient conditions for a Lorentz contact manifold to be a Lorentz Kenmotsu manifold. Moreover, we obtain the optimal Chen first inequality for semi-slant submanifolds in Lorentz Kenmotsu space forms. Furthermore, the equality case of Chen inequality has been discussed.

Chen inequalities on warped product Legendrian submanifolds in Kenmotsu space forms and applications

In the current work, we study the geometry and topology of warped product Legendrian submanifolds in Kenmotsu space forms F2n+1(ϵ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{F}^{2n+1}(\\epsilon )$\\end{document} and derive the first Chen inequality, including extrinsic invariants such as the mean curvature and the length of the warping functions. Additionally, sectional curvature and the δ-invariant are intrinsic invariants related to this inequality. An integral bound is also given in terms of the gradient Ricci curvature for the Bochner operator formula of compact warped product submanifolds. Our primary technique is applying geometry to number structures and solving problems such as problems with Dirichlet eigenvalues.

Chen inequality for general warped product submanifold of Riemannian warped products I×fSm(c)

In the present paper, we investigate the geometry and topology of warped product submanifolds in Riemannian warped product I×fSm(c) and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Our main object is to apply geometrically to number structures and obtained applications of Dirichlet eigenvalues problems. Some new results on mean curvature vanishing are presented, and a partial solution can be found to the well-known problem given by S S Chern. The family of Riemannian warped products generalized to Robertson-Walker space-times

An Invariant of Riemannian Type for Legendrian Warped Product Submanifolds of Sasakian Space Forms

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms D2n+1(ϵ) and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the length of the warping functions. This inequality also involves intrinsic invariants (δ-invariant and sectional curvature). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the gradient Ricci curvature. Some new results on mean curvature vanishing are presented as a partial solution to the well-known problem given by S.S. Chern.

On Statistical Submanifolds in Statistical Manifolds of Quasi-Constant Curvature

We mention some properties of statistical submanifolds in statistical manifolds of quasi-constant curvature. We obtain Chen first inequality and a Chen inequality for the $\delta (2,2)$-invariant for these manifolds.

Recent Developments on the First Chen Inequality in Differential Geometry

One of the most fundamental interests in submanifold theory is to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of submanifolds and find their applications. In this respect, the first author established, in 1993, a basic inequality involving the first δ-invariant, δ(2), and the squared mean curvature of submanifolds in real space forms, known today as the first Chen inequality or Chen’s first inequality. Since then, there have been many papers dealing with this inequality. The purpose of this article is, thus, to present a comprehensive survey on recent developments on this inequality performed by many geometers during the last three decades.

First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

B.-Y. Chen's Inequality for Kähler-like Statistical Submersions

In this paper, we first define the notion of Lagrangian statistical submersion from a K\"ahler-like statistical manifold onto a statistical manifold. Then we prove that the horizontal distribution of a Lagrangian statistical submersion is integrable. Next, we establish Chen-Ricci inequality for Lagrangian statistical submersions from K\"ahler-like statistical manifolds onto statistical manifolds and discuss the equality case of the obtained inequality through a basic tensor introduced by O'Neill that plays the role of the second fundamental form of an isometric immersion. At the end, we give a nontrivial example of a K\"ahler-like statistical submersion.

Improved Chen’s Inequalities for Submanifolds of Generalized Sasakian-Space-Forms

In this article, we derive Chen’s inequalities involving Chen’s δ-invariant δM, Riemannian invariant δ(m1,⋯,mk), Ricci curvature, Riemannian invariant Θk(2≤k≤m), the scalar curvature and the squared of the mean curvature for submanifolds of generalized Sasakian-space-forms endowed with a quarter-symmetric connection. As an application of the obtain inequality, we first derived the Chen inequality for the bi-slant submanifold of generalized Sasakian-space-forms.

Generalized Wintgen Inequality for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

The geometry of Hessian manifolds is a fruitful branch of physics, statistics, Kaehlerian and affine differential geometry. The study of inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature was truly initiated in 2018 by Mihai, A. and Mihai, I. who dealt with Chen-Ricci and Euler inequalities. Later on, Siddiqui, A.N., Ahmad K. and Ozel C. came with the study of Casorati inequality for statistical submanifolds in the same ambient space by using algebraic technique. Also, Chen, B.-Y., Mihai, A. and Mihai, I. obtained a Chen first inequality for such submanifolds. In 2020, Mihai, A. and Mihai, I. studied the Chen inequality for δ(2,2)-invariant. In the development of this topic, we establish the generalized Wintgen inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature. Some examples are also discussed at the end.

Chen inequalities on lightlike hypersurfaces of a GRW spacetime

In this paper, we introduce [Formula: see text]-Ricci curvature and [Formula: see text]-scalar curvature on lightlike hypersurfaces of a GRW spacetime. Using these curvatures, we establish some inequalities for lightlike hypersurfaces of a GRW spacetime. Using these inequalities, we obtain some characterizations on lightlike hypersurfaces. We also get Chen–Ricci inequality and Chen inequality on a screen homothetic lightlike hypersurfaces of a GRW spacetime.

Relations between Extrinsic and Intrinsic Invariants of Statistical Submanifolds in Sasaki-Like Statistical Manifolds

The Chen first inequality and a Chen inequality for the δ(2,2)-invariant on statistical submanifolds of Sasaki-like statistical manifolds, under a curvature condition, are obtained.

A New Algebraic Inequality and Some Applications in Submanifold Theory

We give a simple proof of the Chen inequality involving the Chen invariant δ(k) of submanifolds in Riemannian space forms. We derive Chen’s first inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.

An Algebraic Inequality with Applications to Certain Chen Inequalities

We give a simple proof of the Chen inequality for the Chen invariant δ(2,…,2)︸k terms of submanifolds in Riemannian space forms.

Geometry of Chen invariants in statistical warped product manifolds

In this paper, we derive Chen inequality for statistical submanifold of statistical warped product manifolds [Formula: see text]. Further, we derive Chen inequality for Legendrian statistical submanifold in statistical warped product manifolds [Formula: see text]. We also provide some applications of derived inequalities in a statistical warped product manifold which is equivalent to a hyperbolic space. Moreover, we construct new examples of statistical warped product manifolds to support results.

The δ(2,2)-Invariant on Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature.

We establish Chen inequality for the invariant on statistical submanifolds in Hessian manifolds of constant Hessian curvature. Recently, in co-operation with Chen, we proved a Chen first inequality for such submanifolds. The present authors previously initiated the investigation of statistical submanifolds in Hessian manifolds of constant Hessian curvature; this paper represents a development in this topic.

Chen Inequalities for Statistical Submanifolds of Kähler-Like Statistical Manifolds

We consider Kähler-like statistical manifolds, whose curvature tensor field satisfies a natural condition. For their statistical submanifolds, we prove a Chen first inequality and a Chen inequality for the invariant δ ( 2 , 2 ) .