Editorial

4 pages, 201 KiB

Open AccessEditorial

Differential Geometry and Its Application

by Mića S. Stanković

Axioms 2023, 12(9), 810; https://doi.org/10.3390/axioms12090810 - 23 Aug 2023

Viewed by 1066

We have launched a Special Issue of Axioms which focuses on the generalization of Riemannian spaces and their mappings [...] Full article

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

Research

Jump to: Editorial

11 pages, 267 KiB

Open AccessArticle

On Bochner Flat Kähler B-Manifolds

by Cornelia-Livia Bejan, Galia Nakova and Adara M. Blaga

Axioms 2023, 12(4), 336; https://doi.org/10.3390/axioms12040336 - 30 Mar 2023

Cited by 2 | Viewed by 1230

Abstract

We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only [...] Read more.

We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only if it is a holomorphic Einstein, Bochner flat manifold. Moreover, we provide the necessary and sufficient conditions for a gradient Ricci soliton or a holomorphic

η

-Einstein Kähler manifold with a Norden metric to be Bochner flat. Finally, we show that a Kähler B-manifold is of quasi-constant totally real sectional curvatures if and only if it is a holomorphic

η

-Einstein, Bochner flat manifold. Full article

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

13 pages, 282 KiB

Open AccessArticle

On r-Compactness in Topological and Bitopological Spaces

by Jamal Oudetallah, Rehab Alharbi and Iqbal M. Batiha

Axioms 2023, 12(2), 210; https://doi.org/10.3390/axioms12020210 - 16 Feb 2023

Cited by 5 | Viewed by 1001

Abstract

This paper defines the so-called pairwise r-compactness in topological and bitopological spaces. In particular, several inferred properties of the r-compact spaces and their connections with other topological and bitopological spaces are studied theoretically. As a result, several novel theorems of the [...] Read more.

This paper defines the so-called pairwise r-compactness in topological and bitopological spaces. In particular, several inferred properties of the r-compact spaces and their connections with other topological and bitopological spaces are studied theoretically. As a result, several novel theorems of the r-compact space are generalized on the pairwise r-compact space. The results established in this research paper are new in the field of topology. Full article

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

14 pages, 309 KiB

Open AccessArticle

Certain Curvature Conditions on Kenmotsu Manifolds and 🟉-η-Ricci Solitons

by Halil İbrahim Yoldaş, Abdul Haseeb and Fatemah Mofarreh

Axioms 2023, 12(2), 140; https://doi.org/10.3390/axioms12020140 - 30 Jan 2023

Cited by 5 | Viewed by 1307

Abstract

The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with 🟉-

η

-Ricci solitons. First we find some necessary conditions for such a manifold to be

φ

-Einstein. Then, we study the notion of 🟉-

η

[...] Read more.

The present paper deals with the investigations of a Kenmotsu manifold satisfying certain curvature conditions endowed with 🟉-

η

-Ricci solitons. First we find some necessary conditions for such a manifold to be

φ

-Einstein. Then, we study the notion of 🟉-

η

-Ricci soliton on this manifold and prove some significant results related to this notion. Finally, we construct a nontrivial example of three-dimensional Kenmotsu manifolds to verify some of our results. Full article

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

10 pages, 288 KiB

Open AccessArticle

A Superbundle Description of Differential K-Theory

by Jae Min Lee and Byungdo Park

Axioms 2023, 12(1), 82; https://doi.org/10.3390/axioms12010082 - 12 Jan 2023

Cited by 1 | Viewed by 1104

Abstract

We construct a model of differential K-theory using superbundles with a

Z / 2 Z

-graded connection and a differential form on the base manifold and prove that our model is isomorphic to the Freed–Lott–Klonoff model of differential K-theory. Full article

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

14 pages, 286 KiB

Open AccessArticle

Soft Complete Continuity and Soft Strong Continuity in Soft Topological Spaces

by Samer Al Ghour

Axioms 2023, 12(1), 78; https://doi.org/10.3390/axioms12010078 - 12 Jan 2023

Cited by 5 | Viewed by 1225

Abstract

In this paper, we introduce soft complete continuity as a strong form of soft continuity and we introduce soft strong continuity as a strong form of soft complete continuity. Several characterizations, compositions, and restriction theorems are obtained. Moreover, several preservation theorems regarding soft [...] Read more.

In this paper, we introduce soft complete continuity as a strong form of soft continuity and we introduce soft strong continuity as a strong form of soft complete continuity. Several characterizations, compositions, and restriction theorems are obtained. Moreover, several preservation theorems regarding soft compactness, soft Lindelofness, soft connectedness, soft regularity, soft normality, soft almost regularity, soft mild normality, soft almost compactness, soft almost Lindelofness, soft near compactness, soft near Lindelofness, soft paracompactness, soft near paracompactness, soft almost paracompactness, and soft metacompactness are obtained. In addition to these, the study deals with the correlation between our new concepts in soft topology and their corresponding concepts in general topology; as a result, we show that soft complete continuity (resp. soft strong continuity) in soft topology is an extension of complete continuity (resp. strong continuity) in soft topology. Full article

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

10 pages, 1069 KiB

Open AccessArticle

On a Surface Associated to the Catalan Triangle

by Marilena Jianu, Sever Achimescu, Leonard Dăuş, Ion Mierluş-Mazilu, Adela Mihai and Daniel Tudor

Axioms 2022, 11(12), 685; https://doi.org/10.3390/axioms11120685 - 30 Nov 2022

Cited by 1 | Viewed by 1074

Abstract

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature [...] Read more.

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated. Full article

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

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15 pages, 305 KiB

Open AccessArticle

On h-Quasi-Hemi-Slant Riemannian Maps

by Mohd Bilal, Sushil Kumar, Rajendra Prasad, Abdul Haseeb and Sumeet Kumar

Axioms 2022, 11(11), 641; https://doi.org/10.3390/axioms11110641 - 14 Nov 2022

Cited by 1 | Viewed by 1115

Abstract

In the present article, we indroduce and study h-quasi-hemi-slant (in short, h-qhs) Riemannian maps and almost h-qhs Riemannian maps from almost quaternionic Hermitian manifolds to Riemannian manifolds. We investigate some fundamental results mainly on h-qhs Riemannian maps: the integrability [...] Read more.

In the present article, we indroduce and study h-quasi-hemi-slant (in short, h-qhs) Riemannian maps and almost h-qhs Riemannian maps from almost quaternionic Hermitian manifolds to Riemannian manifolds. We investigate some fundamental results mainly on h-qhs Riemannian maps: the integrability of distributions, geometry of foliations, the condition for such maps to be totally geodesic, etc. At the end of this article, we give two non-trivial examples of this notion. Full article

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

13 pages, 802 KiB

Open AccessArticle

Soft Regular Generalized ω-Closed Sets and Soft ω-T_1/2 Spaces

by Samer Al Ghour

Axioms 2022, 11(10), 529; https://doi.org/10.3390/axioms11100529 - 03 Oct 2022

Cited by 2 | Viewed by 1096

Abstract

Soft

r g ω

-closed sets are introduced as a new class of soft sets that strictly contain the classes of soft

r g

-closed sets and soft

g ω

-closed sets. Furthermore, the behavior of soft

r g ω

-closed sets with [...] Read more.

Soft

r g ω

-closed sets are introduced as a new class of soft sets that strictly contain the classes of soft

r g

-closed sets and soft

g ω

-closed sets. Furthermore, the behavior of soft

r g ω

-closed sets with respect to soft unions, soft intersections, and soft subspaces, as well as induced soft topologies are investigated. Moreover, soft

ω

-

T_{1 / 2}

spaces which is a weaker form soft

T_{1 / 2}

spaces is defined and investigated. In addition to these, the characterizations of soft

r g

-

T_{1 / 2}

spaces and soft

r g ω

-

T_{1 / 2}

spaces are discussed. The work also looks at the relationship between our novel notions in soft topological spaces and their analogs in topological spaces. Full article

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

17 pages, 316 KiB

Open AccessArticle

A Study of Clairaut Semi-Invariant Riemannian Maps from Cosymplectic Manifolds

by Yanlin Li, Rajendra Prasad, Abdul Haseeb, Sushil Kumar and Sumeet Kumar

Axioms 2022, 11(10), 503; https://doi.org/10.3390/axioms11100503 - 26 Sep 2022

Cited by 19 | Viewed by 1542

Abstract

In the present note, we characterize Clairaut semi-invariant Riemannian maps from cosymplectic manifolds to Riemannian manifolds. Moreover, we provide a nontrivial example of such a Riemannian map. Full article

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

17 pages, 326 KiB

Open AccessArticle

Classification of Surfaces of Coordinate Finite Type in the Lorentz–Minkowski 3-Space

by Hassan Al-Zoubi, Alev Kelleci Akbay, Tareq Hamadneh and Mutaz Al-Sabbagh

Axioms 2022, 11(7), 326; https://doi.org/10.3390/axioms11070326 - 04 Jul 2022

Cited by 6 | Viewed by 1257

Abstract

In this paper, we define surfaces of revolution without parabolic points in three-dimensional Lorentz–Minkowski space. Then, we classify this class of surfaces under the condition

Δ^{I I I} x = A x

, where

Δ^{I I I}

is the Laplace operator [...] Read more.

In this paper, we define surfaces of revolution without parabolic points in three-dimensional Lorentz–Minkowski space. Then, we classify this class of surfaces under the condition

Δ^{I I I} x = A x

, where

Δ^{I I I}

is the Laplace operator regarding the third fundamental form, and A is a real square matrix of order 3. We prove that such surfaces are either catenoids or surfaces of Enneper, or pseudo spheres or hyperbolic spaces centered at the origin. Full article

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

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 - 01 Jul 2022

Cited by 20 | Viewed by 1560

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)

15 pages, 315 KiB

Open AccessArticle

Two Invariants for Geometric Mappings

by Nenad O. Vesić, Vladislava M. Milenković and Mića S. Stanković

Axioms 2022, 11(5), 239; https://doi.org/10.3390/axioms11050239 - 20 May 2022

Cited by 1 | Viewed by 1447

Abstract

Two invariants for mappings of affine connection spaces with a special form of deformation tensors are obtained in this paper. We used the methodology of Vesić to obtain the form of these invariants. At the end of this paper, we used these forms [...] Read more.

Two invariants for mappings of affine connection spaces with a special form of deformation tensors are obtained in this paper. We used the methodology of Vesić to obtain the form of these invariants. At the end of this paper, we used these forms to obtain two invariants for third-type almost-geodesic mappings of symmetric affine connection. Full article

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

13 pages, 289 KiB

Open AccessArticle

Soft Rω-Open Sets and the Soft Topology of Soft δ_ω-Open Sets

by Samer Al Ghour

Axioms 2022, 11(4), 177; https://doi.org/10.3390/axioms11040177 - 15 Apr 2022

Cited by 7 | Viewed by 1649

Abstract

The author devotes this paper to defining a new class of soft open sets, namely soft

R ω

-open sets, and investigating their main features. With the help of examples, we show that the class of soft

R ω

-open sets lies strictly [...] Read more.

The author devotes this paper to defining a new class of soft open sets, namely soft

R ω

-open sets, and investigating their main features. With the help of examples, we show that the class of soft

R ω

-open sets lies strictly between the classes of soft regular open sets and soft open sets. We show that soft

R ω

-open subsets of a soft locally countable soft topological space coincide with the soft open sets. Moreover, we show that soft

R ω

-open subsets of a soft anti-locally countable coincide with the soft regular open sets. Moreover, we show that the class of soft

R ω

-open sets is closed under finite soft intersection, and as a conclusion, we show that this class forms a soft base for some soft topology. In addition, we define the soft

δ_{ω}

-closure operator as a new operator in soft topological spaces. Moreover, via the soft

δ_{ω}

-closure operator, we introduce soft

δ_{ω}

-open sets as a new class of soft open sets which form a soft topology. Moreover, we study the correspondence between soft

δ_{ω}

-open in soft topological spaces and

δ_{ω}

-open in topological spaces. Full article

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

12 pages, 297 KiB

Open AccessArticle

Pythagorean Isoparametric Hypersurfaces in Riemannian and Lorentzian Space Forms

by Muhittin Evren Aydın, Adela Mihai and Cihan Özgür

Axioms 2022, 11(2), 59; https://doi.org/10.3390/axioms11020059 - 30 Jan 2022

Cited by 2 | Viewed by 2019

Abstract

We introduce the notion of a Pythagorean hypersurface immersed into an

(n + 1)

-dimensional pseudo-Riemannian space form of constant sectional curvature

c \in \{- 1, 0, 1\}

. By using this definition, we prove in Riemannian setting that if an [...] Read more.

We introduce the notion of a Pythagorean hypersurface immersed into an

(n + 1)

-dimensional pseudo-Riemannian space form of constant sectional curvature

c \in \{- 1, 0, 1\}

. By using this definition, we prove in Riemannian setting that if an isoparametric hypersurface is Pythagorean, then it is totally umbilical with sectional curvature

φ + c,

where

φ

is the Golden Ratio. We also extend this result to Lorentzian ambient space, observing the existence of a non totally umbilical model. Full article

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

6 pages, 351 KiB

Open AccessArticle

A Model of Directed Graph Cofiber

by Zachary McGuirk and Byungdo Park

Axioms 2022, 11(1), 32; https://doi.org/10.3390/axioms11010032 - 16 Jan 2022

Cited by 1 | Viewed by 1900

Abstract

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give [...] Read more.

In the homotopy theory of spaces, the image of a continuous map is contractible to a point in its cofiber. This property does not apply when we discretize spaces and continuous maps to directed graphs and their morphisms. In this paper, we give a construction of a cofiber of a directed graph map whose image is contractible in the cofiber. Our work reveals that a category-theoretically correct construction in continuous setup is no longer correct when it is discretized and hence leads to look at canonical constructions in category theory in a different perspective. Full article

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

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10 pages, 286 KiB

Open AccessArticle

A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings

by Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

Axioms 2021, 10(4), 333; https://doi.org/10.3390/axioms10040333 - 05 Dec 2021

Cited by 2 | Viewed by 2146

Abstract

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented [...] Read more.

This article is devoted to geometrical aspects of conformal mappings of complete Riemannian and Kählerian manifolds and uses the Bochner technique, one of the oldest and most important techniques in modern differential geometry. A feature of this article is that the results presented here are easily obtained using a generalized version of the Bochner technique due to theorems on the connection between the geometry of a complete Riemannian manifold and the global behavior of its subharmonic, superharmonic, and convex functions. Full article

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

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