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16 pages, 271 KiB

Open AccessArticle

On Ricci Solitons and Curvature Properties of Doubly Warped Products with QSMC

by Md Aquib, Vaishali Sah, Sarvesh Kumar Yadav and Jaya Upreti

Axioms 2025, 14(8), 548; https://doi.org/10.3390/axioms14080548 - 22 Jul 2025

Viewed by 148

This paper explores the geometric interplay between the Levi–Civita connection and the quarter-symmetric metric connection on doubly warped product manifolds. We analyze the behavior of Ricci solitons on such manifolds, focusing on the influence of conformal and Killing vector fields within the framework [...] Read more.

This paper explores the geometric interplay between the Levi–Civita connection and the quarter-symmetric metric connection on doubly warped product manifolds. We analyze the behavior of Ricci solitons on such manifolds, focusing on the influence of conformal and Killing vector fields within the framework of quarter-symmetric metric connections (QSMCs). Furthermore, we examine conditions under which the manifold exhibits Einstein properties, presenting new insights into Einstein-like structures in the context of doubly warped product manifolds endowed with a quarter-symmetric metric connection. Full article

(This article belongs to the Special Issue Recent Developments in Differential Geometry and Its Applications)

14 pages, 241 KiB

Open AccessArticle

Analysis of Screen Generic Lightlike Submanifolds in an Indefinite Kaehler Statistical Manifold Endowed with a Quarter-Symmetric Non-Metric Connection

by Vandana Gupta, Jasleen Kaur, Oğuzhan Bahadır and Meraj Ali Khan

Axioms 2025, 14(3), 200; https://doi.org/10.3390/axioms14030200 - 8 Mar 2025

Viewed by 617

Abstract

This paper introduces the notion of screen generic lightlike submanifolds (SGLSs) of an indefinite Kaehler statistical manifold equipped with a quarter-symmetric non-metric (QSNM) connection, supported by suitable illustrations. Assertions for induced connection on the lightlike submanifold and integrability of the distributions are proved. [...] Read more.

This paper introduces the notion of screen generic lightlike submanifolds (SGLSs) of an indefinite Kaehler statistical manifold equipped with a quarter-symmetric non-metric (QSNM) connection, supported by suitable illustrations. Assertions for induced connection on the lightlike submanifold and integrability of the distributions are proved. The characterization theorems on parallelism and geodesicity of the SGLSs are presented. Results for the totally umbilic screen generic lightlike submanifold with a QSNM connection are also established. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

16 pages, 286 KiB

Open AccessArticle

A Study of Generalized Symmetric Metric Connection on Nearly Kenmotsu Manifolds

by Rajesh Kumar, Laltluangkima Chawngthu, Oğuzhan Bahadir and Meraj Ali Khan

Symmetry 2025, 17(3), 317; https://doi.org/10.3390/sym17030317 - 20 Feb 2025

Viewed by 399

Abstract

The focus of this research is on investigating a new category of generalized symmetric metric connections within nearly Kenmotsu manifolds. The study delves into recognizing the generalized symmetric connections of type

(α, β)

, which represent broader versions of [...] Read more.

The focus of this research is on investigating a new category of generalized symmetric metric connections within nearly Kenmotsu manifolds. The study delves into recognizing the generalized symmetric connections of type

(α, β)

, which represent broader versions of the semi-symmetric metric connection

(α = 1, β = 0)

and the quarter-symmetric metric connection

(α = 0, β = 1)

. Full article

(This article belongs to the Section Mathematics)

16 pages, 283 KiB

Open AccessArticle

Geometric Inequalities of Slant Submanifolds in Locally Metallic Product Space Forms

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

Axioms 2024, 13(7), 486; https://doi.org/10.3390/axioms13070486 - 19 Jul 2024

Cited by 8 | Viewed by 990

Abstract

In this particular article, our focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. We specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant [...] Read more.

In this particular article, our focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. We specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant submanifolds. Our investigation takes place within the context of locally metallic product space forms with quarter-symmetric metric connections. Additionally, we delve into the condition that determines when equality is achieved within the inequality. Furthermore, we explore a number of implications of our findings. Full article

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

15 pages, 258 KiB

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

Viewed by 1163

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)

16 pages, 284 KiB

Open AccessArticle

Exploring Conformal Soliton Structures in Tangent Bundles with Ricci-Quarter Symmetric Metric Connections

by Yanlin Li, Aydin Gezer and Erkan Karakas

Mathematics 2024, 12(13), 2101; https://doi.org/10.3390/math12132101 - 4 Jul 2024

Cited by 10 | Viewed by 960

Abstract

In this study, we investigate the tangent bundle

T M

of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection

\tilde{\nabla}

. Our primary goal is to establish the necessary and sufficient conditions for

T M

to exhibit [...] Read more.

In this study, we investigate the tangent bundle

T M

of an n-dimensional (pseudo-)Riemannian manifold M equipped with a Ricci-quarter symmetric metric connection

\tilde{\nabla}

. Our primary goal is to establish the necessary and sufficient conditions for

T M

to exhibit characteristics of various solitons, specifically conformal Yamabe solitons, gradient conformal Yamabe solitons, conformal Ricci solitons, and gradient conformal Ricci solitons. We determine that for

T M

to be a conformal Yamabe soliton, the potential vector field must satisfy certain conditions when lifted vertically, horizontally, or completely from M to

T M

, alongside specific constraints on the conformal factor

λ

and the geometric properties of M. For gradient conformal Yamabe solitons, the conditions involve

λ

and the Hessian of the potential function. Similarly, for

T M

to be a conformal Ricci soliton, we identify conditions involving the lift of the potential vector field, the value of

λ

, and the curvature properties of M. For gradient conformal Ricci solitons, the criteria include the Hessian of the potential function and the Ricci curvature of M. These results enhance the understanding of the geometric properties of tangent bundles under Ricci-quarter symmetric metric connections and provide insights into their transition into various soliton states, contributing significantly to the field of differential geometry. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)

11 pages, 281 KiB

Open AccessArticle

Solitonical Inequality on Submanifolds in Trans-Sasakian Manifolds Coupled with a Slant Factor

by Mohd Danish Siddiqi and Rawan Bossly

Axioms 2024, 13(6), 370; https://doi.org/10.3390/axioms13060370 - 30 May 2024

Viewed by 704

Abstract

In this article, we study the Ricci soliton on slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Moreover, we derive a lower-bound-type inequality for the slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection in terms of gradient [...] Read more.

In this article, we study the Ricci soliton on slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Moreover, we derive a lower-bound-type inequality for the slant submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection in terms of gradient Ricci solitons. We also characterize anti-invariant, invariant, quasi-umbilical submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection for which the same inequality case holds. Finally, we deduce the above inequalities in terms of a scalar concircular field on submanifolds of trans-Sasakian manifolds with a quarter symmetric non-metric connection. Full article

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

16 pages, 290 KiB

Open AccessArticle

Pinching Results for Doubly Warped Products’ Pointwise Bi-Slant Submanifolds in Locally Conformal Almost Cosymplectic Manifolds with a Quarter-Symmetric Connection

by Md Aquib, Ibrahim Al-Dayel, Mohd Aslam, Meraj Ali Khan and Mohammad Shuaib

Symmetry 2024, 16(5), 521; https://doi.org/10.3390/sym16050521 - 25 Apr 2024

Viewed by 1051

Abstract

In this research paper, we establish geometric inequalities that characterize the relationship between the squared mean curvature and the warping functions of a doubly warped product pointwise bi-slant submanifold. Our investigation takes place in the context of locally conformal almost cosymplectic manifolds, which [...] Read more.

In this research paper, we establish geometric inequalities that characterize the relationship between the squared mean curvature and the warping functions of a doubly warped product pointwise bi-slant submanifold. Our investigation takes place in the context of locally conformal almost cosymplectic manifolds, which are equipped with a quarter-symmetric metric connection. We also consider the cases of equality in these inequalities. Additionally, we derive some geometric applications of our obtained results. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 3rd Edition)

15 pages, 300 KiB

Open AccessArticle

Tangent Bundles Endowed with Quarter-Symmetric Non-Metric Connection (QSNMC) in a Lorentzian Para-Sasakian Manifold

by Rajesh Kumar, Lalnunenga Colney, Samesh Shenawy and Nasser Bin Turki

Mathematics 2023, 11(19), 4163; https://doi.org/10.3390/math11194163 - 4 Oct 2023

Cited by 6 | Viewed by 1281

Abstract

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in [...] Read more.

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in an LP-Sasakian manifold to its tangent bundle are investigated. Necessary and sufficient conditions for the lifts of the Ricci tensor to be symmetric and skew-symmetric and the lifts of the projective Ricci tensor to be skew-symmetric in the tangent bundle are given. An example of complete lifts of four-dimensional LP-Sasakian manifolds in the tangent bundle is shown. Full article

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

13 pages, 291 KiB

Open AccessArticle

Certain Results on the Lifts from an LP-Sasakian Manifold to Its Tangent Bundle Associated with a Quarter-Symmetric Metric Connection

by Mohammad Nazrul Islam Khan, Fatemah Mofarreh, Abdul Haseeb and Mohit Saxena

Symmetry 2023, 15(8), 1553; https://doi.org/10.3390/sym15081553 - 8 Aug 2023

Cited by 13 | Viewed by 1610

Abstract

The purpose of this study is to examine the complete lifts from the symmetric and concircular symmetric n-dimensional Lorentzian para-Sasakian manifolds (briefly,

{(L P S)}_{n}

) to its tangent bundle

T M

associated with a Riemannian connection

D^{C}

[...] Read more.

The purpose of this study is to examine the complete lifts from the symmetric and concircular symmetric n-dimensional Lorentzian para-Sasakian manifolds (briefly,

{(L P S)}_{n}

) to its tangent bundle

T M

associated with a Riemannian connection

D^{C}

and a quarter-symmetric metric connection (QSMC)

{\bar{D}}^{C}

. Full article

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

12 pages, 766 KiB

Open AccessArticle

Quarter-Symmetric Metric Connection on a Cosymplectic Manifold

by Miroslav D. Maksimović and Milan Lj. Zlatanović

Mathematics 2023, 11(9), 2209; https://doi.org/10.3390/math11092209 - 8 May 2023

Cited by 1 | Viewed by 1703

Abstract

We study the quarter-symmetric metric A-connection on a cosymplectic manifold. Observing linearly independent curvature tensors with respect to the quarter-symmetric metric A-connection, we construct the Weyl projective curvature tensor on a cosymplectic manifold. In this way, we obtain new conditions for [...] Read more.

We study the quarter-symmetric metric A-connection on a cosymplectic manifold. Observing linearly independent curvature tensors with respect to the quarter-symmetric metric A-connection, we construct the Weyl projective curvature tensor on a cosymplectic manifold. In this way, we obtain new conditions for the manifold to be projectively flat. At the end of the paper, we define

η

-Einstein cosymplectic manifolds of the

θ

-th kind and prove that they coincide with the

η

-Einstein cosymplectic manifold. Full article

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

11 pages, 277 KiB

Open AccessArticle

Tangent Bundles of P-Sasakian Manifolds Endowed with a Quarter-Symmetric Metric Connection

by Mohammad Nazrul Islam Khan, Fatemah Mofarreh and Abdul Haseeb

Symmetry 2023, 15(3), 753; https://doi.org/10.3390/sym15030753 - 19 Mar 2023

Cited by 13 | Viewed by 1950

Abstract

The purpose of this study is to evaluate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection on the tangent bundle

T M

. Certain results on a semisymmetric P-Sasakian manifold, generalized [...] Read more.

The purpose of this study is to evaluate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection on the tangent bundle

T M

. Certain results on a semisymmetric P-Sasakian manifold, generalized recurrent P-Sasakian manifolds, and pseudo-symmetric P-Sasakian manifolds on

T M

are proved. Full article

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

12 pages, 259 KiB

Open AccessArticle

Lifts of a Quarter-Symmetric Metric Connection from a Sasakian Manifold to Its Tangent Bundle

by Mohammad Nazrul Islam Khan, Uday Chand De and Ljubica S. Velimirović

Mathematics 2023, 11(1), 53; https://doi.org/10.3390/math11010053 - 23 Dec 2022

Cited by 24 | Viewed by 1746

Abstract

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle [...] Read more.

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle was established. Some theorems on the curvature tensor and the projective curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection to its tangent bundle were proved. Finally, locally

ϕ

-symmetric Sasakian manifolds with respect to the quarter-symmetric metric connection to its tangent bundle were studied. Full article

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

15 pages, 317 KiB

Open AccessArticle

E-Connections on the ε-Anti-Kähler Manifolds

by Zhizhi Chen, Yanlin Li, Aydin Gezer, Erkan Karakas and Cagri Karaman

Symmetry 2022, 14(9), 1899; https://doi.org/10.3390/sym14091899 - 11 Sep 2022

Viewed by 1575

Abstract

The paper undertakes certain special forms of the quarter symmetric metric and non-metric connections on an

ε

-anti-Kähler manifold. Firstly, we deduce the relation between the Riemannian connection and the special forms of the quarter symmetric metric and non-metric connections. Then, we present [...] Read more.

The paper undertakes certain special forms of the quarter symmetric metric and non-metric connections on an

ε

-anti-Kähler manifold. Firstly, we deduce the relation between the Riemannian connection and the special forms of the quarter symmetric metric and non-metric connections. Then, we present some results concerning the torsion tensors of these connections. In addition, we find the forms of the curvature tensor, the Ricci curvature tensor and scalar curvature of such connections and we search the conditions for the

ε

-anti-Kähler manifold to be an Einstein space with respect to these connections. Finally, we study

U (R i c)

-vector fields with respect to these connections and give some results related to them. Full article

(This article belongs to the Section Mathematics)

16 pages, 329 KiB

Open AccessArticle

Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

by Pengfei Zhang, Yanlin Li, Soumendu Roy and Santu Dey

Symmetry 2021, 13(11), 2189; https://doi.org/10.3390/sym13112189 - 16 Nov 2021

Cited by 9 | Viewed by 2042

Abstract

The outline of this research article is to initiate the development of a ∗-conformal

η

-Ricci–Yamabe soliton in

α

-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of

α

-Cosymplectic manifolds in regard to the quarter-symmetric [...] Read more.

The outline of this research article is to initiate the development of a ∗-conformal

η

-Ricci–Yamabe soliton in

α

-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of

α

-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal

η

-Ricci–Yamabe soliton equation when the potential vector field

ξ

of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional

α

-cosymplectic metric as a ∗-conformal

η

-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results. Full article

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