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15 pages, 279 KiB

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 325

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)

14 pages, 276 KiB

Open AccessArticle

Eigenvalues for the Generalized Laplace Operator of Slant Submanifolds in the Sasakian Space Forms Admitting Semi-Symmetric Metric Connection

by Ibrahim Al-Dayel, Meraj Ali Khan and Sudhakar Kumar Chaubey

Symmetry 2025, 17(2), 279; https://doi.org/10.3390/sym17020279 - 11 Feb 2025

Viewed by 510

Abstract

This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine [...] Read more.

This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine the principal eigenvalue for the generalized-Laplacian operator in closed oriented-slant submanifolds within a Sasakian space form (ssf) with a semi-symmetric metric (ssm) connection. By utilizing our findings on the Laplacian, we extend several Reilly-type inequalities to the generalized Laplacian on slant submanifolds within a unit sphere with a semi-symmetric metric (ssm) connection. The research is concluded with a detailed examination of specific scenarios. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

16 pages, 299 KiB

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 1 | Viewed by 1882

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)

20 pages, 330 KiB

Open AccessArticle

A Note on Nearly Sasakian Manifolds

by Fortuné Massamba and Arthur Nzunogera

Mathematics 2023, 11(12), 2634; https://doi.org/10.3390/math11122634 - 9 Jun 2023

Cited by 3 | Viewed by 1582

Abstract

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. [...] Read more.

A class of nearly Sasakian manifolds is considered in this paper. We discuss the geometric effects of some symmetries on such manifolds and show, under a certain condition, that the class of Ricci semi-symmetric nearly Sasakian manifolds is a subclass of Einstein manifolds. We prove that a Codazzi-type Ricci nearly Sasakian space form is either a Sasakian manifold with a constant

ϕ

-holomorphic sectional curvature

H = 1

or a 5-dimensional proper nearly Sasakian manifold with a constant

ϕ

-holomorphic sectional curvature

H > 1

. We also prove that the spectrum of the operator

H^{2}

generated by the nearly Sasakian space form is a set of a simple eigenvalue of 0 and an eigenvalue of multiplicity 4, and we induce that the underlying space form carries a Sasaki–Einstein structure. We show that there exist integrable distributions with totally geodesic leaves on the same manifolds, and we prove that there are no proper nearly Sasakian space forms with constant sectional curvature. Full article

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

15 pages, 292 KiB

Open AccessArticle

Optimal Inequalities for Submanifolds in Trans-Sasakian Manifolds Endowed with a Semi-Symmetric Metric Connection

by Ion Mihai and Mohammed Mohammed

Symmetry 2023, 15(4), 877; https://doi.org/10.3390/sym15040877 - 6 Apr 2023

Cited by 7 | Viewed by 1406

Abstract

In this paper, we improve the Chen first inequality for special contact slant submanifolds and Legendrian submanifolds, respectively, in (

α, β

) trans-Sasakian generalized Sasakian space forms endowed with a semi-symmetric metric connection. Full article

(This article belongs to the Special Issue Topological Graph Theory and Discrete Geometry II)

13 pages, 305 KiB

Open AccessArticle

Some Basic Inequalities on (ϵ)-Para Sasakian Manifold

by Majid Ali Choudhary, Mohammad Nazrul Islam Khan and Mohd Danish Siddiqi

Symmetry 2022, 14(12), 2585; https://doi.org/10.3390/sym14122585 - 7 Dec 2022

Cited by 6 | Viewed by 1754

Abstract

We propose fundamental inequalities for contact pseudo-slant submanifolds of

(ϵ)

-para Sasakian space form employing generalized normalized

δ

-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a [...] Read more.

We propose fundamental inequalities for contact pseudo-slant submanifolds of

(ϵ)

-para Sasakian space form employing generalized normalized

δ

-Casorati curvature. We characterize submanifolds for which equality cases hold and illustrate the main result with some applications. Further, we have considered a certain type of submanifold for a Ricci soliton and after computing its scalar curvature, developed an inequality to find correlations between intrinsic or extrinsic invariants. Full article

12 pages, 280 KiB

Open AccessArticle

Some Properties of a Concircular Curvature Tensor on Generalized Sasakian-Space-Forms

by Vasant Chavan

AppliedMath 2022, 2(4), 609-620; https://doi.org/10.3390/appliedmath2040035 - 3 Nov 2022

Viewed by 1843

Abstract

The aim of the present paper is to study and investigate the geometrical properties of a concircular curvature tensor on generalized Sasakian-space-forms. In this manner, we obtained results for

ϕ

-concircularly flat,

ϕ

-semisymmetric, locally concircularly symmetric and locally concircularly

ϕ

-symmetric generalized [...] Read more.

The aim of the present paper is to study and investigate the geometrical properties of a concircular curvature tensor on generalized Sasakian-space-forms. In this manner, we obtained results for

ϕ

-concircularly flat,

ϕ

-semisymmetric, locally concircularly symmetric and locally concircularly

ϕ

-symmetric generalized Sasakian-space-forms. Finally, we construct examples of the generalized Sasakian-space-forms to verify some results. Full article

14 pages, 352 KiB

Open AccessArticle

Generalized Lorentzian Sasakian-Space-Forms with

M

-Projective Curvature Tensor

by D. G. Prakasha, M. R. Amruthalakshmi, Fatemah Mofarreh and Abdul Haseeb

Mathematics 2022, 10(16), 2869; https://doi.org/10.3390/math10162869 - 11 Aug 2022

Cited by 3 | Viewed by 1474

Abstract

In this note, the generalized Lorentzian Sasakian-space-form

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfying certain constraints on the

M

-projective curvature tensor

W^{*}

is considered. Here, we characterize the structure [...] Read more.

In this note, the generalized Lorentzian Sasakian-space-form

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfying certain constraints on the

M

-projective curvature tensor

W^{*}

is considered. Here, we characterize the structure

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

when it is, respectively,

M

-projectively flat,

M

-projectively semisymmetric,

M

-projectively pseudosymmetric, and

φ - M

-projectively semisymmetric. Moreover,

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfies the conditions

W^{*} (ζ, V_{1}) \cdot S = 0

,

W^{*} (ζ, V_{1}) \cdot R = 0

and

W^{*} (ζ, V_{1}) \cdot W^{*} = 0

are also examined. Finally, illustrative examples are given for obtained results. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

16 pages, 319 KiB

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 23 | Viewed by 2372

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)

17 pages, 314 KiB

Open AccessArticle

Geometric Classification of Warped Products Isometrically Immersed into Conformal Sasakian Space Froms

by Xiaoming Fan, Yanlin Li, Prince Majeed, Mehraj Ahmad Lone and Sandeep Sharma

Symmetry 2022, 14(3), 608; https://doi.org/10.3390/sym14030608 - 18 Mar 2022

Viewed by 2180

Abstract

Warped products play important roles in differential geometry, general relativity, and symmetry science. In this paper, we study the warped product pointwise semi-slant submanifolds that are isometrically immersed into conformal Sasakian space form. We show that there does not exist any proper warped [...] Read more.

Warped products play important roles in differential geometry, general relativity, and symmetry science. In this paper, we study the warped product pointwise semi-slant submanifolds that are isometrically immersed into conformal Sasakian space form. We show that there does not exist any proper warped product pointwise semi-slant submanifolds in conformal Sasakian manifolds. We derived some geometric inequalities for squared norm of second fundamental form from a warped product pointwise semi-slant submanifold into a conformal Sasakian manifolds. Full article

(This article belongs to the Section Mathematics)

9 pages, 246 KiB

Open AccessArticle

Generalized Sasakian Space Forms Which Are Realized as Real Hypersurfaces in Complex Space Forms

by Alfonso Carriazo, Jong Taek Cho and Verónica Martín-Molina

Mathematics 2020, 8(6), 873; https://doi.org/10.3390/math8060873 - 29 May 2020

Cited by 5 | Viewed by 2168

Abstract

We prove a classification theorem of the generalized Sasakian space forms which are realized as real hypersurfaces in complex space forms. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications)

15 pages, 280 KiB

Open AccessArticle

A Closed Form for Slant Submanifolds of Generalized Sasakian Space Forms

by Pablo Alegre, Joaquín Barrera and Alfonso Carriazo

Mathematics 2019, 7(12), 1238; https://doi.org/10.3390/math7121238 - 13 Dec 2019

Cited by 1 | Viewed by 2327

Abstract

The Maslov form is a closed form for a Lagrangian submanifold of

C^{m}

, and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the [...] Read more.

The Maslov form is a closed form for a Lagrangian submanifold of

C^{m}

, and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal. Full article

(This article belongs to the Special Issue Inequalities in Geometry and Applications)

20 pages, 830 KiB

Open AccessArticle

The First Fundamental Equation and Generalized Wintgen-Type Inequalities for Submanifolds in Generalized Space Forms

by Mohd. Aquib, Michel Nguiffo Boyom, Mohammad Hasan Shahid and Gabriel-Eduard Vîlcu

Mathematics 2019, 7(12), 1151; https://doi.org/10.3390/math7121151 - 1 Dec 2019

Cited by 13 | Viewed by 3139

Abstract

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some [...] Read more.

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces. Full article

(This article belongs to the Special Issue Inequalities in Geometry and Applications)

15 pages, 232 KiB

Open AccessArticle

Characterizations of the Total Space (Indefinite Trans-Sasakian Manifolds) Admitting a Semi-Symmetric Metric Connection

by Dae Ho Jin and Jae Won Lee

Axioms 2018, 7(3), 68; https://doi.org/10.3390/axioms7030068 - 10 Sep 2018

Viewed by 3388

Abstract

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric [...] Read more.

We investigate recurrent, Lie-recurrent, and Hopf lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a semi-symmetric metric connection. In these hypersurfaces, we obtain several new results. Moreover, we characterize that the total space (an indefinite generalized Sasakian space form) with a semi-symmetric metric connection is an indefinite Kenmotsu space form under various lightlike hypersurfaces. Full article

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

10 pages, 235 KiB

Open AccessArticle

Optimal Inequalities for the Casorati Curvatures of Submanifolds in Generalized Space Forms Endowed with Semi-Symmetric Non-Metric Connections

by Guoqing He, Hairong Liu and Liang Zhang

Symmetry 2016, 8(11), 113; https://doi.org/10.3390/sym8110113 - 27 Oct 2016

Cited by 19 | Viewed by 3921

Abstract

In this paper, we prove some optimal inequalities involving the intrinsic scalar curvature and the extrinsic Casorati curvature of submanifolds in a generalized complex space form with a semi-symmetric non-metric connection and a generalized Sasakian space form with a semi-symmetric non-metric connection. Moreover, [...] Read more.

In this paper, we prove some optimal inequalities involving the intrinsic scalar curvature and the extrinsic Casorati curvature of submanifolds in a generalized complex space form with a semi-symmetric non-metric connection and a generalized Sasakian space form with a semi-symmetric non-metric connection. Moreover, we show that in both cases, the equalities hold if and only if submanifolds are invariantly quasi-umbilical. Full article

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