Mathematics

Editorial

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6 pages, 177 KiB

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

Viewed by 218

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)

Research

Jump to: Editorial, Review, Other

11 pages, 265 KiB

Open AccessFeature PaperArticle

Ricci–Bourguignon Almost Solitons with Vertical Torse-Forming Potential on Almost Contact Complex Riemannian Manifolds

by Mancho Manev

Mathematics 2025, 13(2), 243; https://doi.org/10.3390/math13020243 - 13 Jan 2025

Viewed by 487

Abstract

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with Ricci–Bourguignon-like almost solitons. These almost solitons are a generalization of the known Ricci–Bourguignon almost solitons, in which, in addition to the main metric, the associated metric of the [...] Read more.

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are equipped with Ricci–Bourguignon-like almost solitons. These almost solitons are a generalization of the known Ricci–Bourguignon almost solitons, in which, in addition to the main metric, the associated metric of the manifold is also involved. In the present paper, the soliton potential is specialized to be pointwise collinear with the Reeb vector field of the manifold structure, as well as torse-forming with respect to the two Levi-Civita connections of the pair of B-metrics. The forms of the Ricci tensor and the scalar curvatures generated by the pair of B-metrics on the studied manifolds with the additional structures have been found. In the three-dimensional case, an explicit example is constructed and some of the properties obtained in the theoretical part are illustrated. Full article

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

18 pages, 300 KiB

Open AccessArticle

Magnetic Curves in Homothetic s-th Sasakian Manifolds

by Şaban Güvenç and Cihan Özgür

Mathematics 2025, 13(1), 159; https://doi.org/10.3390/math13010159 - 4 Jan 2025

Viewed by 553

Abstract

We investigate normal magnetic curves in

(2 n + s)

-dimensional homothetic s-th Sasakian manifolds as a generalization of S-manifolds. We show that a curve

γ

is a normal magnetic curve in a homothetic s-th Sasakian manifold if [...] Read more.

We investigate normal magnetic curves in

(2 n + s)

-dimensional homothetic s-th Sasakian manifolds as a generalization of S-manifolds. We show that a curve

γ

is a normal magnetic curve in a homothetic s-th Sasakian manifold if and only if its osculating order satisfies

r \leq 3

and it belongs to a family of

θ_{i}

-slant helices. Additionally, we construct a homothetic s-th Sasakian manifold using generalized D-homothetic transformations and present the parametric equations of normal magnetic curves in this manifold. Full article

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

8 pages, 298 KiB

Open AccessArticle

Infinite Dimensional Maximal Torus Revisited

by Mohamed Lemine H. Bouleryah, Akram Ali and Piscoran Laurian-Ioan

Mathematics 2024, 12(23), 3829; https://doi.org/10.3390/math12233829 - 4 Dec 2024

Viewed by 662

Abstract

Let

T^{m}

be the maximal torus of a set of

m \times m

unitary diagonal matrices. Let

T

be a collection of all maps that rigidly rotate every circle of latitude of the sphere with a fixed angle.

T

is also a [...] Read more.

Let

T^{m}

be the maximal torus of a set of

m \times m

unitary diagonal matrices. Let

T

be a collection of all maps that rigidly rotate every circle of latitude of the sphere with a fixed angle.

T

is also a maximal torus, and we shall prove in this paper that

T

is the topological limit inf of

T^{m}

. Full article

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

21 pages, 336 KiB

Open AccessFeature PaperArticle

Twistor and Reflector Spaces for Paraquaternionic Contact Manifolds

by Stefan Ivanov, Ivan Minchev and Marina Tchomakova

Mathematics 2024, 12(21), 3355; https://doi.org/10.3390/math12213355 - 25 Oct 2024

Viewed by 777

Abstract

We consider certain fiber bundles over paraquaternionic contact manifolds, called twistor and reflector spaces. We show that the twistor space carries an integrable CR structure (Cauchy–Riemann structure) and the reflector space is an integrable para-CR structure, both with neutral signatures. Full article

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

14 pages, 305 KiB

Open AccessArticle

Rigidity and Triviality of Gradient r-Almost Newton-Ricci-Yamabe Solitons

by Mohd Danish Siddiqi and Fatemah Mofarreh

Mathematics 2024, 12(20), 3173; https://doi.org/10.3390/math12203173 - 10 Oct 2024

Viewed by 710

Abstract

In this paper, we develop the concept of gradient r-Almost Newton-Ricci-Yamabe solitons (in brief, gradient r-ANRY solitons) immersed in a Riemannian manifold. We deduce the minimal and totally geodesic criteria for the hypersurface of a Riemannian manifold in terms of the [...] Read more.

In this paper, we develop the concept of gradient r-Almost Newton-Ricci-Yamabe solitons (in brief, gradient r-ANRY solitons) immersed in a Riemannian manifold. We deduce the minimal and totally geodesic criteria for the hypersurface of a Riemannian manifold in terms of the gradient r-ANRY soliton. We also exhibit a Schur-type inequality and discuss the triviality of the gradient r-ANRY soliton in the case of a compact manifold. Finally, we demonstrate the completeness and noncompactness of the r-Newton-Ricci-Yamabe soliton on the hypersurface of the Riemannian manifold. Full article

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

15 pages, 281 KiB

Open AccessArticle

Some Properties of the Potential Field of an Almost Ricci Soliton

by Adara M. Blaga and Sharief Deshmukh

Mathematics 2024, 12(19), 3049; https://doi.org/10.3390/math12193049 - 28 Sep 2024

Cited by 1 | Viewed by 903

Abstract

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral [...] Read more.

In this article, we are interested in finding necessary and sufficient conditions for a compact almost Ricci soliton to be a trivial Ricci soliton. As a first result, we show that positive Ricci curvature and, for a nonzero constant c, the integral of

Ric (c ξ, c ξ)

satisfying a generic inequality on an n-dimensional compact and connected almost Ricci soliton

(M^{n}, g, ξ, σ)

are necessary and sufficient conditions for it to be isometric to the n-sphere

S^{n} (c)

. As another result, we show that, if the affinity tensor of the soliton vector field

ξ

vanishes and the scalar curvature

τ

of an n-dimensional compact almost Ricci soliton

(M^{n}, g, ξ, σ)

satisfies

τ (n σ - τ) \geq 0

, then

(M^{n}, g, ξ, σ)

is a trivial Ricci soliton. Finally, on an n-dimensional compact almost Ricci soliton

(M^{n}, g, ξ, σ)

, we consider the Hodge decomposition

ξ = \bar{ξ} + \nabla h

, where

div \bar{ξ} = 0

, and we use the bound on the integral of

Ric (\bar{ξ}, \bar{ξ})

and an integral inequality involving the scalar curvature to find another characterization of the n-sphere. Full article

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

17 pages, 306 KiB

Open AccessArticle

On LP-Kenmotsu Manifold with Regard to Generalized Symmetric Metric Connection of Type (α, β)

by Doddabhadrappla Gowda Prakasha, Nasser Bin Turki, Mathad Veerabhadraswamy Deepika and İnan Ünal

Mathematics 2024, 12(18), 2915; https://doi.org/10.3390/math12182915 - 19 Sep 2024

Cited by 1 | Viewed by 807

Abstract

In the current article, we examine Lorentzian para-Kenmotsu (shortly, LP-Kenmotsu) manifolds with regard to the generalized symmetric metric connection

\nabla^{G}

of type

(α, β)

. First, we obtain the expressions for curvature tensor, Ricci tensor and scalar curvature of [...] Read more.

In the current article, we examine Lorentzian para-Kenmotsu (shortly, LP-Kenmotsu) manifolds with regard to the generalized symmetric metric connection

\nabla^{G}

of type

(α, β)

. First, we obtain the expressions for curvature tensor, Ricci tensor and scalar curvature of an LP-Kenmotsu manifold with regard to the connection

\nabla^{G}

. Next, we analyze LP-Kenmotsu manifolds equipped with the connection

\nabla^{G}

that are locally symmetric, Ricci semi-symmetric, and

φ

-Ricci symmetric and also demonstrated that in all these situations the manifold is an Einstein one with regard to the connection

\nabla^{G}

. Moreover, we obtain some conclusions about projectively flat, projectively semi-symmetric and

φ

-projectively flat LP-Kenmotsu manifolds concerning the connection

\nabla^{G}

along with several consequences through corollaries. Ultimately, we provide a 5-dimensional LP-Kenmotsu manifold example to validate the derived expressions. Full article

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

12 pages, 258 KiB

Open AccessArticle

Projective Vector Fields on Semi-Riemannian Manifolds

by Norah Alshehri and Mohammed Guediri

Mathematics 2024, 12(18), 2914; https://doi.org/10.3390/math12182914 - 19 Sep 2024

Cited by 1 | Viewed by 691 | Correction

Abstract

This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field P on such a manifold is also a conformal vector field with potential function

ψ

and the vector field

ζ

dual [...] Read more.

This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field P on such a manifold is also a conformal vector field with potential function

ψ

and the vector field

ζ

dual to

d ψ

does not change its causal character, then P is homothetic, or

ζ

is a light-like vector field. Additionally, it is shown that a complete Riemannian manifold admits a projective vector field that is also conformal and non-Killing if and only if it is locally Euclidean. The paper also presents other results related to the characterization of Killing and parallel vector fields using the Ricci curvature and the Hessian of the function given by the inner product of the vector field. Full article

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

14 pages, 266 KiB

Open AccessArticle

Investigating Statistical Submersions and Their Properties

by Aliya Naaz Siddiqui and Fatimah Alghamdi

Mathematics 2024, 12(17), 2750; https://doi.org/10.3390/math12172750 - 5 Sep 2024

Viewed by 1222

Abstract

This research aims to prove a sharp relationship between statistical submersions and doubly minimal immersions. We consider a non-trivial statistical submersion on a statistical manifold with isometric fibers. Then, we investigate that it cannot be isometrically immersed as a doubly minimal manifold into [...] Read more.

This research aims to prove a sharp relationship between statistical submersions and doubly minimal immersions. We consider a non-trivial statistical submersion on a statistical manifold with isometric fibers. Then, we investigate that it cannot be isometrically immersed as a doubly minimal manifold into any statistical manifold of non-positive sectional curvature using a submersion invariant for dual connections and additional conditions. Full article

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

14 pages, 271 KiB

Open AccessArticle

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

by Hanan Alohali, Sharief Deshmukh, Bang-Yen Chen and Hemangi Madhusudan Shah

Mathematics 2024, 12(17), 2628; https://doi.org/10.3390/math12172628 - 24 Aug 2024

Cited by 1 | Viewed by 897

Abstract

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and [...] Read more.

For a compact Riemannian m-manifold

(M^{m}, g), m > 1,

endowed with a nontrivial conformal vector field

ζ

with a conformal factor

σ

, there is an associated skew-symmetric tensor

φ

called the associated tensor, and also,

ζ

admits the Hodge decomposition

ζ = \bar{ζ} + \nabla ρ

, where

\bar{ζ}

satisfies

div \bar{ζ} = 0

, which is called the Hodge vector, and

ρ

is the Hodge potential of

ζ

. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on

M^{m}

. The first result of this article states that a compact Riemannian m-manifold

M^{m}

is an m-sphere

S^{m} (c)

if and only if (1) for a nonzero constant c, the function

- σ / c

is a solution of the Poisson equation

Δ ρ = m σ

, and (2) the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

. The second result states that if

M^{m}

has constant scalar curvature

τ = m (m - 1) c > 0

, then it is an

S^{m} (c)

if and only if the Ricci curvature satisfies

R i c (\bar{ζ}, \bar{ζ}) \geq {∥φ∥}^{2}

and the Hodge potential

ρ

satisfies a certain static perfect fluid equation. The third result provides another new characterization of

S^{m} (c)

using the affinity tensor of the Hodge vector

\bar{ζ}

of a conformal vector field

ζ

on a compact Riemannian manifold

M^{m}

with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold

M^{m}

,

m > 2

, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field

ζ

whose affinity tensor vanishes identically and

ζ

annihilates its associated tensor

φ

. Full article

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

14 pages, 292 KiB

Open AccessArticle

Invariants for Second Type Almost Geodesic Mappings of Symmetric Affine Connection Space

by Nenad O. Vesić, Dušan J. Simjanović and Branislav M. Randjelović

Mathematics 2024, 12(15), 2329; https://doi.org/10.3390/math12152329 - 25 Jul 2024

Viewed by 739

Abstract

This paper presents the results concerning a space of invariants for second type almost geodesic mappings. After discussing the general formulas of invariants for mappings of symmetric affine connection spaces, based on these formulas, invariants for second type almost geodesic mappings of symmetric [...] Read more.

This paper presents the results concerning a space of invariants for second type almost geodesic mappings. After discussing the general formulas of invariants for mappings of symmetric affine connection spaces, based on these formulas, invariants for second type almost geodesic mappings of symmetric affine connection spaces and Riemannian spaces are obtained, as well as their mutual connection. Also, one invariant of Thomas type and two invariants of Weyl type for almost geodesic mappings of the second type were attained. Full article

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

9 pages, 253 KiB

Open AccessArticle

The de Rham Cohomology Classes of Hemi-Slant Submanifolds in Locally Product Riemannian Manifolds

by Mustafa Gök and Erol Kılıç

Mathematics 2024, 12(11), 1730; https://doi.org/10.3390/math12111730 - 2 Jun 2024

Viewed by 689

Abstract

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham [...] Read more.

This paper aims to discuss the de Rham cohomology of hemi-slant submanifolds in locally product Riemannian manifolds. The integrability and geodesical invariance conditions of the distributions derived from the definition of a hemi-slant submanifold are given. The existence and non-triviality of de Rham cohomology classes of hemi-slant submanifolds are investigated. Finally, an example is presented. Full article

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

16 pages, 7502 KiB

Open AccessArticle

Parameterizations of Delaunay Surfaces from Scratch

by Clementina D. Mladenova and Ivaïlo M. Mladenov

Mathematics 2024, 12(10), 1570; https://doi.org/10.3390/math12101570 - 17 May 2024

Viewed by 1251

Abstract

Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one [...] Read more.

Starting with the most fundamental differential-geometric principles we derive here new explicit parameterizations of the Delaunay surfaces of revolution which depend on two real parameters with fixed ranges. Besides, we have proved that these parameters have very clear geometrical meanings. The first one is responsible for the size of the surface under consideration and the second one specifies its shape. Depending on the concrete ranges of these parameters any of the Delaunay surfaces which is neither a cylinder nor the plane is classified unambiguously either as a first or a second kind Delaunay surface. According to this classification spheres are Delaunay surfaces of first kind while the catenoids are Delaunay surfaces of second kind. Geometry of both classes is established meaning that the coefficients of their fundamental forms are found in explicit form. Full article

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

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25 pages, 309 KiB

Open AccessArticle

On the Structure of SO(3): Trace and Canonical Decompositions

by Demeter Krupka and Ján Brajerčík

Mathematics 2024, 12(10), 1490; https://doi.org/10.3390/math12101490 - 10 May 2024

Cited by 2 | Viewed by 973

Abstract

This paper is devoted to some selected topics of the theory of special orthogonal group SO(3). First, we discuss the trace of orthogonal matrices and its relation to the characteristic polynomial; on this basis, the partition of SO(3) formed by conjugation classes is [...] Read more.

This paper is devoted to some selected topics of the theory of special orthogonal group SO(3). First, we discuss the trace of orthogonal matrices and its relation to the characteristic polynomial; on this basis, the partition of SO(3) formed by conjugation classes is described by trace mapping. Second, we show that every special orthogonal matrix can be expressed as the product of three elementary special orthogonal matrices. Explicit formulas for the decomposition as needed for applications in differential geometry and physics as symmetry transformations are given. Full article

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

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 1188

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

Cited by 1 | Viewed by 990

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 11 | Viewed by 1520

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 1391

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 17 | Viewed by 1374

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 PaperEditor’s ChoiceArticle

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

Cited by 1 | Viewed by 1362

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 1066

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

Cited by 1 | Viewed by 1217

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 5 | Viewed by 1436

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 - 6 Oct 2023

Cited by 1 | Viewed by 1261

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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Figure 1

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

Cited by 2 | Viewed by 1343

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

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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 - 6 Oct 2023

Cited by 1 | Viewed by 1749

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)

Other

Jump to: Editorial, Research, Review

1 pages, 156 KiB

Open AccessCorrection

Correction: Alshehri, N.; Guediri, M. Projective Vector Fields on Semi-Riemannian Manifolds. Mathematics 2024, 12, 2914

by Norah Alshehri and Mohammed Guediri

Mathematics 2024, 12(24), 3958; https://doi.org/10.3390/math12243958 - 17 Dec 2024

Viewed by 360

Abstract

In our recent paper [1], we stated in Theorem 10 that on an n-dimensional semi-Riemannian manifold

(N, h)

with

n \geq 2

, if P is a projective vector field that is also conformal, satisfying [...] Read more.

In our recent paper [1], we stated in Theorem 10 that on an n-dimensional semi-Riemannian manifold

(N, h)

with

n \geq 2

, if P is a projective vector field that is also conformal, satisfying

£_{P} h = 2 ψ h

, and the vector field

ζ

, dual to

d ψ

, maintains a consistent causal character, then either P is homothetic or

ζ

is light-like [...] Full article

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

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