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16 pages, 325 KiB

Open AccessArticle

Statistical Solitonic Impact on Submanifolds of Kenmotsu Statistical Manifolds

by Abdullah Ali H. Ahmadini, Mohd. Danish Siddiqi and Aliya Naaz Siddiqui

Mathematics 2024, 12(9), 1279; https://doi.org/10.3390/math12091279 - 24 Apr 2024

Viewed by 297

Abstract

In this article, we delve into the study of statistical solitons on submanifolds of Kenmotsu statistical manifolds, introducing the presence of concircular vector fields. This investigation is further extended to study the behavior of almost quasi-Yamabe solitons on submanifolds with both concircular and [...] Read more.

In this article, we delve into the study of statistical solitons on submanifolds of Kenmotsu statistical manifolds, introducing the presence of concircular vector fields. This investigation is further extended to study the behavior of almost quasi-Yamabe solitons on submanifolds with both concircular and concurrent vector fields. Concluding our research, we offer a compelling example featuring a 5-dimensional Kenmotsu statistical manifold that accommodates both a statistical soliton and an almost quasi-Yamabe soliton. This example serves to reinforce and validate the principles discussed throughout our study. Full article

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

14 pages, 277 KiB

Open AccessArticle

Lifts of a Semi-Symmetric Metric Connection from Sasakian Statistical Manifolds to Tangent Bundle

by Rajesh Kumar, Sameh Shenawy, Nasser Bin Turki, Lalnunenga Colney and Uday Chand De

Mathematics 2024, 12(2), 226; https://doi.org/10.3390/math12020226 - 10 Jan 2024

Viewed by 525

Abstract

The lifts of Sasakian statistical manifolds associated with a semi-symmetric metric connection in the tangent bundle are characterized in the current research. The relationship between the complete lifts of a statistical manifold with semi-symmetric metric connections and Sasakian statistical manifolds with a semi-symmetric [...] Read more.

The lifts of Sasakian statistical manifolds associated with a semi-symmetric metric connection in the tangent bundle are characterized in the current research. The relationship between the complete lifts of a statistical manifold with semi-symmetric metric connections and Sasakian statistical manifolds with a semi-symmetric metric connection in the tangent bundle is investigated. We also discuss the classification of Sasakian statistical manifolds with respect to semi-symmetric metric connections in the tangent bundle. Finally, we derive an example of the lifts of Sasakian statistical manifolds to the tangent bundle. Full article

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

14 pages, 294 KiB

Open AccessArticle

Killing and 2-Killing Vector Fields on Doubly Warped Products

by Adara M. Blaga and Cihan Özgür

Mathematics 2023, 11(24), 4983; https://doi.org/10.3390/math11244983 - 17 Dec 2023

Cited by 1 | Viewed by 673

Abstract

We provide a condition for a 2-Killing vector field on a compact Riemannian manifold to be Killing and apply the result to doubly warped product manifolds. We establish a connection between the property of a vector field on a doubly warped product manifold [...] Read more.

We provide a condition for a 2-Killing vector field on a compact Riemannian manifold to be Killing and apply the result to doubly warped product manifolds. We establish a connection between the property of a vector field on a doubly warped product manifold and its components on the factor manifolds to be Killing or 2-Killing. We also prove that a Killing vector field on the doubly warped product gives rise to a Ricci soliton factor manifold if and only if it is an Einstein manifold. If a component of a Killing vector field on the doubly warped product is of a gradient type, then, under certain conditions, the corresponding factor manifold is isometric to the Euclidean space. Moreover, we provide necessary and sufficient conditions for a doubly warped product to reduce to a direct product. As applications, we characterize the 2-Killing vector fields on the doubly warped spacetimes, particularly on the standard static spacetime and on the generalized Robertson–Walker spacetime. Full article

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

15 pages, 298 KiB

Open AccessArticle

Eigenvectors of the De-Rham Operator

by Nasser Bin Turki, Sharief Deshmukh and Gabriel-Eduard Vîlcu

Mathematics 2023, 11(24), 4942; https://doi.org/10.3390/math11244942 - 13 Dec 2023

Viewed by 891

Abstract

We aim to examine the influence of the existence of a nonzero eigenvector

ζ

of the de-Rham operator

Γ

on a k-dimensional Riemannian manifold

(N^{k}, g)

. If the vector

ζ

annihilates the de-Rham operator, such a vector [...] Read more.

We aim to examine the influence of the existence of a nonzero eigenvector

ζ

of the de-Rham operator

Γ

on a k-dimensional Riemannian manifold

(N^{k}, g)

. If the vector

ζ

annihilates the de-Rham operator, such a vector field is called a de-Rham harmonic vector field. It is shown that for each nonzero vector field

ζ

on

(N^{k}, g)

, there are two operators

T_{ζ}

and

Ψ_{ζ}

associated with

ζ

, called the basic operator and the associated operator of

ζ

, respectively. We show that the existence of an eigenvector

ζ

of

Γ

on a compact manifold

(N^{k}, g)

, such that the integral of

Ric (ζ, ζ)

admits a certain lower bound, forces

(N^{k}, g)

to be isometric to a k-dimensional sphere. Moreover, we prove that the existence of a de-Rham harmonic vector field

ζ

on a connected and complete Riemannian space

(N^{k}, g)

, having

div (ζ) \neq 0

and annihilating the associated operator

Ψ_{ζ}

, forces

(N^{k}, g)

to be isometric to the k-dimensional Euclidean space, provided that the squared length of the covariant derivative of

ζ

possesses a certain lower bound. Full article

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

14 pages, 327 KiB

Open AccessArticle

Bounds for Eigenvalues of q-Laplacian on Contact Submanifolds of Sasakian Space Forms

by Yanlin Li, Fatemah Mofarreh, Abimbola Abolarinwa, Norah Alshehri and Akram Ali

Mathematics 2023, 11(23), 4717; https://doi.org/10.3390/math11234717 - 21 Nov 2023

Cited by 6 | Viewed by 664

Abstract

This study establishes new upper bounds for the mean curvature and constant sectional curvature on Riemannian manifolds for the first positive eigenvalue of the q-Laplacian. In particular, various estimates are provided for the first eigenvalue of the q-Laplace operator on closed [...] Read more.

This study establishes new upper bounds for the mean curvature and constant sectional curvature on Riemannian manifolds for the first positive eigenvalue of the q-Laplacian. In particular, various estimates are provided for the first eigenvalue of the q-Laplace operator on closed orientated

(l + 1)

-dimensional special contact slant submanifolds in a Sasakian space form,

{\tilde{M}}^{2 k + 1} (ϵ)

, with a constant

ψ_{1}

-sectional curvature,

ϵ

. From our main results, we recovered the Reilly-type inequalities, which were proven before this study. Full article

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

22 pages, 408 KiB

Open AccessFeature PaperArticle

On Some Weingarten Surfaces in the Special Linear Group SL(2,

R

)

by Marian Ioan Munteanu

Mathematics 2023, 11(22), 4636; https://doi.org/10.3390/math11224636 - 13 Nov 2023

Viewed by 738

Abstract

We classify Weingarten conoids in the real special linear group

SL (2, R)

. In particular, there is no linear Weingarten nontrivial conoids in

SL (2, R)

. We also prove that the only conoids in [...] Read more.

We classify Weingarten conoids in the real special linear group

SL (2, R)

. In particular, there is no linear Weingarten nontrivial conoids in

SL (2, R)

. We also prove that the only conoids in

SL (2, R)

with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup

N

is a Weingarten surface. Full article

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

6 pages, 235 KiB

Open AccessFeature PaperArticle

Categorical Join and Generating Families in Diffeological Spaces

by E. Macías and R. Mehrabi

Mathematics 2023, 11(21), 4503; https://doi.org/10.3390/math11214503 - 31 Oct 2023

Cited by 1 | Viewed by 591

Abstract

We prove that a diffeological space is diffeomorphic to the categorical join of any generating family of plots. Full article

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

14 pages, 326 KiB

Open AccessArticle

Solitons Equipped with a Semi-Symmetric Metric Connection with Some Applications on Number Theory

by Ali H. Hakami, Mohd. Danish Siddiqi, Aliya Naaz Siddiqui and Kamran Ahmad

Mathematics 2023, 11(21), 4452; https://doi.org/10.3390/math11214452 - 27 Oct 2023

Viewed by 692

Abstract

A solution to an evolution equation that evolves along symmetries of the equation is called a self-similar solution or soliton. In this manuscript, we present a study of

η

-Ricci solitons (

η

-RS) for an interesting manifold called the

(ε)

[...] Read more.

A solution to an evolution equation that evolves along symmetries of the equation is called a self-similar solution or soliton. In this manuscript, we present a study of

η

-Ricci solitons (

η

-RS) for an interesting manifold called the

(ε)

-Kenmotsu manifold (

(ε)

-

K M

), endowed with a semi-symmetric metric connection (briefly, a SSM-connection). We discuss Ricci and

η

-Ricci solitons with a SSM-connection satisfying certain curvature restrictions. In addition, we consider the characteristics of the gradient

η

-Ricci solitons (a special case of

η

-Ricci soliton), with a Poisson equation on the same ambient manifold for a SSM-connection. In addition, we derive an inequality for the lower bound of gradient

η

-Ricci solitons for

(ε)

-Kenmotsu manifold, with a semi-symmetric metric connection. Finally, we explore a number theoretic approach in the form of Pontrygin numbers to the

(ε)

-Kenmotsu manifold equipped with a semi-symmetric metric connection. Full article

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

10 pages, 282 KiB

Open AccessArticle

Weak Nearly Sasakian and Weak Nearly Cosymplectic Manifolds

by Vladimir Rovenski

Mathematics 2023, 11(20), 4377; https://doi.org/10.3390/math11204377 - 21 Oct 2023

Cited by 1 | Viewed by 760

Abstract

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak [...] Read more.

Weak contact metric structures on a smooth manifold, introduced by V. Rovenski and R. Wolak in 2022, have provided new insight into the theory of classical structures. In this paper, we define new structures of this kind (called weak nearly Sasakian and weak nearly cosymplectic and nearly Kähler structures), study their geometry and give applications to Killing vector fields. We introduce weak nearly Kähler manifolds (generalizing nearly Kähler manifolds), characterize weak nearly Sasakian and weak nearly cosymplectic hypersurfaces in such Riemannian manifolds and prove that a weak nearly cosymplectic manifold with parallel Reeb vector field is locally the Riemannian product of a real line and a weak nearly Kähler manifold. Full article

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

13 pages, 393 KiB

Open AccessArticle

Vertices of Ovals with Constant Width Relative to Particular Circles

by Adel Al-rabtah and Kamal Al-Banawi

Mathematics 2023, 11(19), 4179; https://doi.org/10.3390/math11194179 - 06 Oct 2023

Viewed by 695

Abstract

In this article, we study ovals of constant width in a plane, comparing them to particular circles. We use the vertices on the oval, after counting them, as a reference to measure the length of the curve between opposite points. A new proof [...] Read more.

In this article, we study ovals of constant width in a plane, comparing them to particular circles. We use the vertices on the oval, after counting them, as a reference to measure the length of the curve between opposite points. A new proof of Barbier’s theorem is introduced. A distance function from the origin to the points of the oval is introduced, and it is shown that extreme values of the distance function occur at the vertices and opposite points. Comparisons are made between ovals and particular circles. We prove that the differences in the distances from the origin between the particular circles and the ovals are small and within a certain range. We also prove that all types of ovals described in this paper are analytically and geometrically enclosed between two defined circles centered at the origin. Full article

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

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10 pages, 236 KiB

Open AccessArticle

Norden Golden Manifolds with Constant Sectional Curvature and Their Submanifolds

by Fulya Şahin, Bayram Şahin and Feyza Esra Erdoğan

Mathematics 2023, 11(15), 3301; https://doi.org/10.3390/math11153301 - 27 Jul 2023

Viewed by 668

Abstract

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if a Norden golden manifold has a constant real sectional curvature, the manifold is flat. For this reason, the notions of holomorphic-like sectional curvature and holomorphic-like [...] Read more.

This paper discusses the Norden golden manifold having a constant sectional curvature. First, it is shown that if a Norden golden manifold has a constant real sectional curvature, the manifold is flat. For this reason, the notions of holomorphic-like sectional curvature and holomorphic-like bisectional curvature on the Norden golden manifold are investigated, but it is seen that these notions do not work on the Norden golden manifold. This shows the need for a new concept of sectional curvature. In this direction, a new notion of sectional curvature (Norden golden sectional curvature) is proposed, an example is given, and if this new sectional curvature is constant, the curvature tensor field of the Norden golden manifold is expressed in terms of the metric tensor field. Since the geometry of the submanifolds of manifolds with constant sectional curvature has nice properties, the last section of this paper examines the semi-invariant submanifolds of the Norden golden space form. Full article

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

Review

Jump to: Research

50 pages, 653 KiB

Open AccessReview

Recent Developments on the First Chen Inequality in Differential Geometry

by Bang-Yen Chen and Gabriel-Eduard Vîlcu

Mathematics 2023, 11(19), 4186; https://doi.org/10.3390/math11194186 - 06 Oct 2023

Viewed by 907

Abstract

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 [...] Read more.

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. Full article

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

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