Chen Inequalities Research Articles

Pointwise hemi-slant Riemannian maps ($\mathcal{PHSRM}$) from almost Hermitian manifolds

In 2022, the notion of pointwise slant Riemannian maps were introduced by Y. G\"{u}nd\"{u}zalp and M. A. Akyol [Journal of Geometry and Physics, 179, 104589, 2022] as a natural generalization of slant Riemannian maps \cite{s3}, slant Riemannian submersions \cite{lee}, slant submanifolds \cite{c1}. As a generalization of pointwise slant Riemannian maps and many subclasses notions (see also: \cite{ga2}), we introduce {\textit pointwise hemi-slant Riemannian maps (briefly, $\mathcal{PHSRM}$)} from almost Hermitian manifolds to Riemannian manifolds, giving non-trivial (proper) examples and investigate some properties of the map, we deal with their properties: the J-pluriharmonicity, the J-invariant, and the totally geodesicness of the map. Finally, we study some curvature relations in complex space form, involving Chen inequalities and Casorati curvatures for pointwise hemi-slant Riemannian maps, respectively.

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.

Some Chen Inequalities for Submanifolds in Trans-Sasakian Manifolds Admitting a Semi-Symmetric Non-Metric Connection

In the present article, we study submanifolds tangent to the Reeb vector field in trans-Sasakian manifolds. We prove Chen’s first inequality and the Chen–Ricci inequality, respectively, for such submanifolds in trans-Sasakian manifolds which admit a semi-symmetric non-metric connection. Moreover, a generalized Euler inequality for special contact slant submanifolds in trans-Sasakian manifolds endowed with a semi-symmetric non-metric connection is obtained.

Basic Chen inequalities for statistical submanifolds of cosymplectic statistical space forms

Basic Chen inequalities for statistical submanifolds of cosymplectic statistical space forms

Chen Inequalities for Isotropic Submanifolds in Pseudo-Riemannian Space Forms

The class of isotropic submanifolds in pseudo-Riemannian manifolds is a distinguished family of submanifolds; they have been studied by several authors. In this article we establish Chen inequalities for isotropic immersions. An example of an isotropic immersion for which the equality case in the Chen first inequality holds is given.

Ricci and scalar curvature inequalities for semi-slant Riemannian submersions in complex space forms

This paper is devoted to the obtain Chen inequalities containing the Ricci and scalar inequalities on the vertical and the horizontal distributions for semi-slant Riemannian submersions from complex space forms onto Riemannian manifolds. The equality case of the obtained inequalities is discussed. At the end, some geometric consequences are obtained. Moreover, many examples are constructed.

Optimal inequalities for submanifolds of Riemann manifolds of nearly quasi-constant curvature with quarter-symmetric connection

In this paper, we obtain Chen inequalities for submanifolds in Riemannian manifolds of nearly quasi-constant curvature with a special kind of quartersymmetric connection and discuss the equality case of the inequalities. We also obtain some Casorati inequalities for submanifolds in Riemannian manifolds of nearly quasi-constant curvature with the quarter-symmetric connection.

General Chen Inequalities for Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature

Chen’s first inequality for statistical submanifolds in Hessian manifolds of constant Hessian curvature was obtained by B.-Y. Chen et al. Other particular cases of Chen inequalities in a statistical setting were given by different authors. The objective of the present article is to establish the general Chen inequalities for statistical submanifolds in Hessian manifolds of constant Hessian curvature.

Chen inequalities on spacelike hypersurfaces of a GRW spacetime

In this paper, we establish Chen-Ricci inequality and Chen inequalities for spacelike hypersurfaces in GRW spacetimes. Considering these inequalities, we give some characterizations. Equality cases are also considered. We show that M is a totally geodesic or a totally umbilical in case of equalities. Moreover, we find some basic inequalities for shape operator of such hypersurfaces and obtain some inequalities for spacelike slices in GRW spacetimes.

Chen Inequalities for Spacelike Submanifolds in Statistical Manifolds of Type Para-Kähler Space Forms

In this paper, we prove some inequalities between intrinsic and extrinsic curvature invariants, namely involving the Chen first invariant and the mean curvature of totally real and holomorphic spacelike submanifolds in statistical manifolds of type para-Kähler space forms. Furthermore, we investigate the equality cases of these inequalities. As illustrations of the applications of the above inequalities, we consider a few examples.

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.

Chen inequalities for statistical submersions between statistical manifolds

We study statistical submersions between statistical manifolds. In particular, we establish Chen–Ricci inequalities of statistical submersions between statistical manifolds and a [Formula: see text] Chen-type inequality for statistical submersions. Some applications are also given.

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.

δ(2,2)-Invariant for Lagrangian Submanifolds in Quaternionic Space Forms

In the geometry of submanifolds, Chen inequalities represent one of the most important tool to find relationships between intrinsic and extrinsic invariants; the aim is to find sharp such inequalities. In this paper we establish an optimal inequality for the Chen invariant δ ( 2 , 2 ) on Lagrangian submanifolds in quaternionic space forms, regarded as a problem of constrained maxima.

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 ) .

Chen inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connections

In this paper, we obtain Chen’s inequalities for submanifolds of complex space forms and Sasakian space forms with quarter-symmetric connections.

Chen Inequalities for Submanifolds of Real Space Forms with a Ricci Quarter-Symmetric Metric Connection

In this paper, we establish some inequalities for submanifolds of real space forms endowed with a Ricci quarter-symmetric metric connection. Using these inequalities, we obtain the relation between Ricci curvature, scalar curvature and the mean curvature endowed with the Ricci quartersymmetric metric connection.

Chen Inequalities on Lightlike Hypersurface of a Lorentzian Manifold with Semi-Symmetric Metric Connection

Chen Inequalities on Lightlike Hypersurface of a Lorentzian Manifold with Semi-Symmetric Metric Connection

$\delta^{\sharp}(2,2)$-Ideal Centroaffine Hypersurfaces of Dimension $5$

The notion of an ideal submanifold was introduced by Chen at the end of the last century. A survey of recent results in this area can be found in his book [9]. Recently, in [10], an optimal collection of Chen's inequalities was obtained for Lagrangian submanifolds in complex space forms. As shown in [2], these inequalities have an immediate counterpart in centroaffine differential geometry. Centroaffine hypersurfaces realising the equality in one of these inequalities are called ideal centroaffine hypersurfaces. So far, most results in this area have only been related with $3$- and $4$-dimensional $\delta^{\sharp}(2)$-ideal centroaffine hypersurfaces. The purpose of this paper is to classify $\delta^{\sharp}(2,2)$-ideal hypersurfaces of dimension~$5$ in centroaffine differential geometry.

Chen Inequalities For Submanifolds Of Generalized Space Forms With a Semi-symmetric Metric Connection

We investigate sharp inequalities for submanifolds in both generalized complex space forms and generalized Sasakian space forms with a semisymmetric metric connection.