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8 pages, 217 KB

Open AccessArticle

A Simple Proof of Chen–Ricci Inequality and Applications

by Ion Mihai

Geometry 2025, 2(4), 20; https://doi.org/10.3390/geometry2040020 - 1 Dec 2025

Viewed by 419

In the present paper, we give a simple proof of the Chen–Ricci inequality for submanifolds in Riemannian and Lorentzian space forms, respectively. Moreover, we extend the Chen–Ricci inequality to submanifolds in Lorentzian manifolds with a semi-symmetric non-metric connection. Full article

15 pages, 279 KB

Open AccessArticle

Slant Submersions in Generalized Sasakian Space Forms and Some Optimal Inequalities

by Md Aquib

Axioms 2025, 14(6), 417; https://doi.org/10.3390/axioms14060417 - 29 May 2025

Viewed by 621

Abstract

This research examines key inequalities associated with the scalar and Ricci curvatures of slant submersions within generalized Sasakian space forms (GSSFs). We establish significant geometric constraints and conduct a detailed analysis of the conditions that lead to equality in these bounds. By expanding [...] Read more.

This research examines key inequalities associated with the scalar and Ricci curvatures of slant submersions within generalized Sasakian space forms (GSSFs). We establish significant geometric constraints and conduct a detailed analysis of the conditions that lead to equality in these bounds. By expanding the existing framework of curvature inequalities, our results provide new insights into the geometric characteristics of slant submersions in contact structures. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

6 pages, 177 KB

Open AccessEditorial

Differentiable Manifolds and Geometric Structures

by Adara M. Blaga

Mathematics 2025, 13(7), 1082; https://doi.org/10.3390/math13071082 - 26 Mar 2025

Cited by 2 | Viewed by 780

Abstract

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the [...] Read more.

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing) [...] Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

13 pages, 259 KB

Open AccessArticle

Classification Results of f-Biharmonic Immersion in T-Space Forms

by Md Aquib, Mohd Iqbal and Sarvesh Kumar Yadav

Axioms 2025, 14(4), 242; https://doi.org/10.3390/axioms14040242 - 22 Mar 2025

Viewed by 480

Abstract

We investigate f-biharmonic submanifolds in T-space form, where we analyze different scenarios and provide necessary and sufficient conditions for f-biharmonicity. We also derive a non-existence result for f-biharmonic submanifolds where

ξ

and

ϕ Ω

are tangents. Finally, we derive the Chen–Ricci inequality for [...] Read more.

We investigate f-biharmonic submanifolds in T-space form, where we analyze different scenarios and provide necessary and sufficient conditions for f-biharmonicity. We also derive a non-existence result for f-biharmonic submanifolds where

ξ

and

ϕ Ω

are tangents. Finally, we derive the Chen–Ricci inequality for submanifolds of T-space forms and provide the conditions under which this inequality becomes equality. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

18 pages, 274 KB

Open AccessArticle

Quaternion Statistical Submanifolds and Submersions

by Aliya Naaz Siddiqui and Fatimah Alghamdi

Mathematics 2025, 13(1), 53; https://doi.org/10.3390/math13010053 - 27 Dec 2024

Viewed by 919

Abstract

This paper aims to develop a general theory of quaternion Kahlerian statistical manifolds and to study quaternion CR-statistical submanifolds in such ambient manifolds. It extends the existing theories of quaternion submanifolds and totally real submanifolds. Additionally, the work examines quaternion Kahlerian statistical submersions, [...] Read more.

This paper aims to develop a general theory of quaternion Kahlerian statistical manifolds and to study quaternion CR-statistical submanifolds in such ambient manifolds. It extends the existing theories of quaternion submanifolds and totally real submanifolds. Additionally, the work examines quaternion Kahlerian statistical submersions, including illustrative examples. The exploration also includes an analysis of the total space and fibers under certain conditions with example(s) in support. Moreover, Chen–Ricci inequality on the vertical distribution is derived for quaternion Kahlerian statistical submersions from quaternion Kahlerian statistical manifolds. Full article

(This article belongs to the Section B: Geometry and Topology)

18 pages, 285 KB

Open AccessArticle

Chen-like Inequalities for Submanifolds in Kähler Manifolds Admitting Semi-Symmetric Non-Metric Connections

by Ion Mihai and Andreea Olteanu

Symmetry 2024, 16(10), 1401; https://doi.org/10.3390/sym16101401 - 21 Oct 2024

Cited by 1 | Viewed by 1530

Abstract

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such [...] Read more.

The geometry of submanifolds in Kähler manifolds is an important research topic. In the present paper, we study submanifolds in complex space forms admitting a semi-symmetric non-metric connection. We prove the Chen–Ricci inequality, Chen basic inequality, and a generalized Euler inequality for such submanifolds. These inequalities provide estimations of the mean curvature (the main extrinsic invariants) in terms of intrinsic invariants: Ricci curvature, the Chen invariant, and scalar curvature. In the proofs, we use the sectional curvature of a semi-symmetric, non-metric connection recently defined by A. Mihai and the first author, as well as its properties. Full article

(This article belongs to the Special Issue Symmetry in Metric Spaces and Topology)

15 pages, 258 KB

Open AccessArticle

Quarter-Symmetric Non-Metric Connection of Non-Integrable Distributions

by Shuo Chen and Haiming Liu

Symmetry 2024, 16(7), 848; https://doi.org/10.3390/sym16070848 - 5 Jul 2024

Cited by 1 | Viewed by 1344

Abstract

In this paper, we focus on non-integrable distributions with a quarter-symmetric non-metric connection (QSNMC) in generalized Riemannian manifold. First, by studying a quarter-symmetric connection on the generalized Riemannian manifold, we obtain the condition that the connection is non-metric. Then, the Gauss, Codazzi and [...] Read more.

In this paper, we focus on non-integrable distributions with a quarter-symmetric non-metric connection (QSNMC) in generalized Riemannian manifold. First, by studying a quarter-symmetric connection on the generalized Riemannian manifold, we obtain the condition that the connection is non-metric. Then, the Gauss, Codazzi and Ricci equations are proved for non-integrable distributions with respect to a quarter-symmetric non-metric connection in generalized Riemannian manifold. Furthermore, we deduce Chen’s inequalities for non-integrable distributions of real space forms with a quarter-symmetric non-metric connection in generalized Riemannian manifold as applications. After that, we give some examples of non-integrable distributions in Riemannian manifold with quarter-symmetric non-metric connection. Full article

(This article belongs to the Section Mathematics)

13 pages, 263 KB

Open AccessArticle

Analyzing the Ricci Tensor for Slant Submanifolds in Locally Metallic Product Space Forms with a Semi-Symmetric Metric Connection

by Yanlin Li, Md Aquib, Meraj Ali Khan, Ibrahim Al-Dayel and Khalid Masood

Axioms 2024, 13(7), 454; https://doi.org/10.3390/axioms13070454 - 4 Jul 2024

Cited by 8 | Viewed by 1223

Abstract

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the [...] Read more.

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with

n = 2

, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

16 pages, 299 KB

Open AccessArticle

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

by Mohammed Mohammed, Fortuné Massamba, Ion Mihai, Abd Elmotaleb A. M. A. Elamin and M. Saif Aldien

Axioms 2024, 13(3), 195; https://doi.org/10.3390/axioms13030195 - 15 Mar 2024

Cited by 2 | Viewed by 2064

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

13 pages, 270 KB

Open AccessArticle

Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms

by Yanlin Li, Meraj Ali Khan, MD Aquib, Ibrahim Al-Dayel and Maged Zakaria Youssef

Axioms 2024, 13(3), 183; https://doi.org/10.3390/axioms13030183 - 11 Mar 2024

Cited by 11 | Viewed by 1930

Abstract

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic [...] Read more.

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

20 pages, 345 KB

Open AccessArticle

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

by Fatemah Abdullah Alghamdi, Lamia Saeed Alqahtani, Ali H. Alkhaldi and Akram Ali

Mathematics 2023, 11(23), 4718; https://doi.org/10.3390/math11234718 - 21 Nov 2023

Cited by 2 | Viewed by 1271

Abstract

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms

D^{2 n + 1} (ϵ)

and obtain the first Chen inequality that involves extrinsic invariants like the mean curvature and the [...] Read more.

In the present paper, we investigate the geometry and topology of warped product Legendrian submanifolds in Sasakian space forms

D^{2 n + 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. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Submanifolds)

18 pages, 343 KB

Open AccessArticle

Optimal Inequalities for Hemi-Slant Riemannian Submersions

by Mehmet Akif Akyol, Ramazan Demir, Nergiz Önen Poyraz and Gabriel-Eduard Vîlcu

Mathematics 2022, 10(21), 3993; https://doi.org/10.3390/math10213993 - 27 Oct 2022

Cited by 4 | Viewed by 1811

Abstract

In the present paper, we establish some basic inequalities involving the Ricci and scalar curvature of the vertical and the horizontal distributions for hemi-slant submersions having the total space a complex space form. We also discuss the equality case of the obtained inequalities [...] Read more.

In the present paper, we establish some basic inequalities involving the Ricci and scalar curvature of the vertical and the horizontal distributions for hemi-slant submersions having the total space a complex space form. We also discuss the equality case of the obtained inequalities and provide illustrative examples. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Submanifolds)

16 pages, 319 KB

Open AccessArticle

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

by Yanlin Li, Mohan Khatri, Jay Prakash Singh and Sudhakar K. Chaubey

Axioms 2022, 11(7), 324; https://doi.org/10.3390/axioms11070324 - 1 Jul 2022

Cited by 24 | Viewed by 2656

Abstract

In this article, we derive Chen’s inequalities involving Chen’s

δ

-invariant

δ_{M}

, Riemannian invariant

δ (m_{1}, \dots, m_{k})

, Ricci curvature, Riemannian invariant

Θ_{k} (2 \leq k \leq m)

, the scalar [...] Read more.

In this article, we derive Chen’s inequalities involving Chen’s

δ

-invariant

δ_{M}

, Riemannian invariant

δ (m_{1}, \dots, m_{k})

, Ricci curvature, Riemannian invariant

Θ_{k} (2 \leq k \leq 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. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application)

10 pages, 263 KB

Open AccessArticle

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

by Aliya Naaz Siddiqui, Ali Hussain Alkhaldi and Lamia Saeed Alqahtani

Mathematics 2022, 10(10), 1727; https://doi.org/10.3390/math10101727 - 18 May 2022

Cited by 5 | Viewed by 1630

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

10 pages, 247 KB

Open AccessArticle

A New Algebraic Inequality and Some Applications in Submanifold Theory

by Ion Mihai and Radu-Ioan Mihai

Mathematics 2021, 9(11), 1175; https://doi.org/10.3390/math9111175 - 23 May 2021

Cited by 3 | Viewed by 1950

Abstract

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. Full article

(This article belongs to the Special Issue Applications of Inequalities and Functional Analysis)

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