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4 pages, 175 KiB

Open AccessEditorial

Differential Geometry and Its Application, 2nd Edition

by Mića S. Stanković

Axioms 2024, 13(8), 561; https://doi.org/10.3390/axioms13080561 - 18 Aug 2024

Viewed by 573

Abstract

With this Editorial, we present a Special Issue of Axioms entitled ‘Differential Geometry and Its Application, 2nd edition’ [...] Full article

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

Research

Jump to: Editorial

18 pages, 3650 KiB

Open AccessArticle

Ensuring Topological Data-Structure Preservation under Autoencoder Compression Due to Latent Space Regularization in Gauss–Legendre Nodes

by Chethan Krishnamurthy Ramanaik, Anna Willmann, Juan-Esteban Suarez Cardona, Pia Hanfeld, Nico Hoffmann and Michael Hecht

Axioms 2024, 13(8), 535; https://doi.org/10.3390/axioms13080535 - 7 Aug 2024

Viewed by 793

Abstract

We formulate a data-independent latent space regularization constraint for general unsupervised autoencoders. The regularization relies on sampling the autoencoder Jacobian at Legendre nodes, which are the centers of the Gauss–Legendre quadrature. Revisiting this classic allows us to prove that regularized autoencoders ensure a [...] Read more.

We formulate a data-independent latent space regularization constraint for general unsupervised autoencoders. The regularization relies on sampling the autoencoder Jacobian at Legendre nodes, which are the centers of the Gauss–Legendre quadrature. Revisiting this classic allows us to prove that regularized autoencoders ensure a one-to-one re-embedding of the initial data manifold into its latent representation. Demonstrations show that previously proposed regularization strategies, such as contractive autoencoding, cause topological defects even in simple examples, as do convolutional-based (variational) autoencoders. In contrast, topological preservation is ensured by standard multilayer perceptron neural networks when regularized using our approach. This observation extends from the classic FashionMNIST dataset to (low-resolution) MRI brain scans, suggesting that reliable low-dimensional representations of complex high-dimensional datasets can be achieved using this regularization technique. Full article

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

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11 pages, 255 KiB

Open AccessArticle

Quasi-Canonical Biholomorphically Projective Mappings of Generalized Riemannian Space in the Eisenhart Sense

by Vladislava M. Milenković and Mića S. Stanković

Axioms 2024, 13(8), 528; https://doi.org/10.3390/axioms13080528 - 3 Aug 2024

Viewed by 500

Abstract

In this paper, quasi-canonical biholomorphically projective and equitorsion quasi-canonical biholomorphically projective mappings are defined. Some relations between the corresponding curvature tensors of the generalized Riemannian spaces

G R_{N}

and

G {\bar{R}}_{N}

are obtained. At the end, the invariant geometric object [...] Read more.

In this paper, quasi-canonical biholomorphically projective and equitorsion quasi-canonical biholomorphically projective mappings are defined. Some relations between the corresponding curvature tensors of the generalized Riemannian spaces

G R_{N}

and

G {\bar{R}}_{N}

are obtained. At the end, the invariant geometric object of an equitorsion quasi-canonical biholomorphically projective mapping is found. Full article

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

7 pages, 241 KiB

Open AccessArticle

Geometry of Torsion Gerbes and Flat Twisted Vector Bundles

by Byungdo Park

Axioms 2024, 13(8), 504; https://doi.org/10.3390/axioms13080504 - 26 Jul 2024

Viewed by 393

Abstract

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle [...] Read more.

Gerbes and higher gerbes are geometric cocycles representing higher degree cohomology classes, and are attracting considerable interest in differential geometry and mathematical physics. We prove that a 2-gerbe has a torsion Dixmier–Douady class if and only if the gerbe has locally constant cocycle data. As an application, we give an alternative description of flat twisted vector bundles in terms of locally constant transition maps. These results generalize to n-gerbes for

n = 1

and

n \geq 3

, providing insights into the structure of higher gerbes and their applications to the geometry of twisted vector bundles. Full article

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

16 pages, 283 KiB

Open AccessArticle

Geometric Inequalities of Slant Submanifolds in Locally Metallic Product Space Forms

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

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

Cited by 6 | Viewed by 510

Abstract

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

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

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

13 pages, 263 KiB

Open AccessArticle

Analyzing the Ricci Tensor for Slant Submanifolds in Locally Metallic Product Space Forms with a Semi-Symmetric Metric Connection

by Yanlin Li, Md Aquib, Meraj Ali Khan, Ibrahim Al-Dayel and Khalid Masood

Axioms 2024, 13(7), 454; https://doi.org/10.3390/axioms13070454 - 4 Jul 2024

Cited by 5 | Viewed by 445

Abstract

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the [...] Read more.

This article explores the Ricci tensor of slant submanifolds within locally metallic product space forms equipped with a semi-symmetric metric connection (SSMC). Our investigation includes the derivation of the Chen–Ricci inequality and an in-depth analysis of its equality case. More precisely, if the mean curvature vector at a point vanishes, then the equality case of this inequality is achieved by a unit tangent vector at the point if and only if the vector belongs to the normal space. Finally, we have shown that when a point is a totally geodesic point or is totally umbilical with

n = 2

, the equality case of this inequality holds true for all unit tangent vectors at the point, and conversely. Full article

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

20 pages, 298 KiB

Open AccessArticle

Cross Curvature Solitons of Lorentzian Three-Dimensional Lie Groups

by Shahroud Azami, Mehdi Jafari, Abdul Haseeb and Abdullah Ali H. Ahmadini

Axioms 2024, 13(4), 211; https://doi.org/10.3390/axioms13040211 - 25 Mar 2024

Cited by 1 | Viewed by 1323

Abstract

In this paper, we study left-invariant cross curvature solitons on Lorentzian three-dimensional Lie groups and classify these solitons. Full article

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

Viewed by 1474

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)

13 pages, 270 KiB

Open AccessArticle

Chen–Ricci Inequality for Isotropic Submanifolds in Locally Metallic Product Space Forms

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

Axioms 2024, 13(3), 183; https://doi.org/10.3390/axioms13030183 - 11 Mar 2024

Cited by 9 | Viewed by 1219

Abstract

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic [...] Read more.

In this article, we study isotropic submanifolds in locally metallic product space forms. Firstly, we establish the Chen–Ricci inequality for such submanifolds and determine the conditions under which the inequality becomes equality. Additionally, we explore the minimality of Lagrangian submanifolds in locally metallic product space forms, and we apply the result to create a classification theorem for isotropic submanifolds whose mean curvature is constant. More specifically, we have demonstrated that the submanifolds are either a product of two Einstein manifolds with Einstein constants, or they are isometric to a totally geodesic submanifold. To support our findings, we provide several examples. Full article

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

17 pages, 348 KiB

Open AccessArticle

Differential Cohomology and Gerbes: An Introduction to Higher Differential Geometry

by Byungdo Park

Axioms 2024, 13(1), 60; https://doi.org/10.3390/axioms13010060 - 19 Jan 2024

Viewed by 1164

Abstract

Differential cohomology is a topic that has been attracting considerable interest. Many interesting applications in mathematics and physics have been known, including the description of WZW terms, string structures, the study of conformal immersions, and classifications of Ramond–Ramond fields, to list a few. [...] Read more.

Differential cohomology is a topic that has been attracting considerable interest. Many interesting applications in mathematics and physics have been known, including the description of WZW terms, string structures, the study of conformal immersions, and classifications of Ramond–Ramond fields, to list a few. Additionally, it is an interesting application of the theory of infinity categories. In this paper, we give an expository account of differential cohomology and the classification of higher line bundles (also known as

S^{1}

-banded gerbes) with a connection.We begin with how Čech cohomology is used to classify principal bundles and define their characteristic classes, introduce differential cohomology à la Cheeger and Simons, and introduce

S^{1}

-banded gerbes with a connection. Full article

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

16 pages, 571 KiB

Open AccessArticle

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

by M. K. Gupta, Abha Sahu, C. K. Yadav, Anjali Goswami and Chetan Swarup

Axioms 2023, 12(12), 1133; https://doi.org/10.3390/axioms12121133 - 18 Dec 2023

Cited by 1 | Viewed by 1451

Abstract

The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system [...] Read more.

The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. In this approach, we describe the evolution of the Oregonator model in geometric terms, by considering it as a geodesic in a Finsler space. We have found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. We have obtained the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points. Further, a comprehensive examination was conducted to compare the linear stability and Jacobi stability of the Oregonator model at its equilibrium points, and We highlight these instances with a few illustrative examples. Full article

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

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14 pages, 514 KiB

Open AccessArticle

Surface Pencil Couple with Bertrand Couple as Joint Principal Curves in Galilean 3-Space

by Nadia Alluhaibi and Rashad A. Abdel-Baky

Axioms 2023, 12(11), 1022; https://doi.org/10.3390/axioms12111022 - 30 Oct 2023

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Abstract

A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of [...] Read more.

A principal curve on a surface plays a paramount role in reasonable implementations. A curve on a surface is a principal curve if its tangents are principal directions. Using the Serret–Frenet frame, the surface pencil couple can be expressed as linear combinations of the components of the local frames in Galilean 3-space

G_{3}

. With these parametric representations, a family of surfaces using principal curves (curvature lines) are constructed, and the necessary and sufficient condition for the given Bertrand couple to be the principal curves on these surfaces are derived in our approach. Moreover, the necessary and sufficient condition for the given Bertrand couple to satisfy the principal curves and the geodesic requirements are also analyzed. As implementations of our main consequences, we expound upon some models to confirm the method. Full article

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

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18 pages, 787 KiB

Open AccessArticle

Kinematic Geometry of a Timelike Line Trajectory in Hyperbolic Locomotions

by Areej A. Almoneef and Rashad A. Abdel-Baky

Axioms 2023, 12(10), 915; https://doi.org/10.3390/axioms12100915 - 26 Sep 2023

Viewed by 806

Abstract

This study utilizes the axodes invariants to derive novel hyperbolic proofs of the Euler–Savary and Disteli formulae. The inflection circle, which is widely recognized, is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical [...] Read more.

This study utilizes the axodes invariants to derive novel hyperbolic proofs of the Euler–Savary and Disteli formulae. The inflection circle, which is widely recognized, is situated on the hyperbolic dual unit sphere, in accordance with the principles of the kinematic theory of spherical locomotions. Subsequently, a timelike line congruence is defined and its spatial equivalence is thoroughly studied. The formulated assertions degenerate into a quadratic form, which facilitates a comprehensive understanding of the geometric features of the inflection line congruence. Full article

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

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11 pages, 313 KiB

Open AccessArticle

A Solitonic Study of Riemannian Manifolds Equipped with a Semi-Symmetric Metric ξ-Connection

by Abdul Haseeb, Sudhakar Kumar Chaubey, Fatemah Mofarreh and Abdullah Ali H. Ahmadini

Axioms 2023, 12(9), 809; https://doi.org/10.3390/axioms12090809 - 23 Aug 2023

Cited by 3 | Viewed by 981

Abstract

The aim of this paper is to characterize a Riemannian 3-manifold

M^{3}

equipped with a semi-symmetric metric

ξ

-connection

\tilde{\nabla}

with

ρ

-Einstein and gradient

ρ

-Einstein solitons. The existence of a gradient

ρ

-Einstein soliton in an

M^{3}

admitting [...] Read more.

The aim of this paper is to characterize a Riemannian 3-manifold

M^{3}

equipped with a semi-symmetric metric

ξ

-connection

\tilde{\nabla}

with

ρ

-Einstein and gradient

ρ

-Einstein solitons. The existence of a gradient

ρ

-Einstein soliton in an

M^{3}

admitting

\tilde{\nabla}

is ensured by constructing a non-trivial example, and hence some of our results are verified. By using standard tensorial technique, we prove that the scalar curvature of

(M^{3}, \tilde{\nabla})

satisfies the Poisson equation

Δ R = \frac{4 (2 - σ - 6 ρ)}{ρ}

. Full article

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

17 pages, 356 KiB

Open AccessArticle

η-Ricci–Yamabe Solitons along Riemannian Submersions

by Mohd Danish Siddiqi, Fatemah Mofarreh, Mehmet Akif Akyol and Ali H. Hakami

Axioms 2023, 12(8), 796; https://doi.org/10.3390/axioms12080796 - 17 Aug 2023

Cited by 1 | Viewed by 1127

Abstract

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the

η

-Ricci–Yamabe soliton (

η

-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an

η

-RY soliton, an [...] Read more.

In this paper, we investigate the geometrical axioms of Riemannian submersions in the context of the

η

-Ricci–Yamabe soliton (

η

-RY soliton) with a potential field. We give the categorization of each fiber of Riemannian submersion as an

η

-RY soliton, an

η

-Ricci soliton, and an

η

-Yamabe soliton. Additionally, we consider the many circumstances under which a target manifold of Riemannian submersion is an

η

-RY soliton, an

η

-Ricci soliton, an

η

-Yamabe soliton, or a quasi-Yamabe soliton. We deduce a Poisson equation on a Riemannian submersion in a specific scenario if the potential vector field

ω

of the soliton is of gradient type =:grad

(γ)

and provide some examples of an

η

-RY soliton, which illustrates our finding. Finally, we explore a number theoretic approach to Riemannian submersion with totally geodesic fibers. Full article

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

11 pages, 260 KiB

Open AccessArticle

Finsler Warped Product Metrics with Special Curvature Properties

by Lingen Sun, Xiaoling Zhang and Mengke Wu

Axioms 2023, 12(8), 784; https://doi.org/10.3390/axioms12080784 - 12 Aug 2023

Viewed by 818

Abstract

The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space–time. In this paper, we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in [...] Read more.

The class of warped product metrics can often be interpreted as key space models for the general theory of relativity and theory of space–time. In this paper, we study several non-Riemannian quantities in Finsler geometry. These non-Riemannian quantities play an important role in understanding the geometric properties of Finsler metrics. In particular, we find differential equations of Finsler warped product metrics with vanishing

χ

-curvature or vanishing H-curvature. Furthermore, we show that, for Finsler warped product metrics, the

χ

-curvature vanishes if and only if the H-curvature vanishes. Full article

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

18 pages, 326 KiB

Open AccessArticle

Infinitesimal Affine Transformations and Mutual Curvatures on Statistical Manifolds and Their Tangent Bundles

by Esmaeil Peyghan, Davood Seifipour and Ion Mihai

Axioms 2023, 12(7), 667; https://doi.org/10.3390/axioms12070667 - 6 Jul 2023

Cited by 2 | Viewed by 1058

Abstract

The purpose of this paper is to find some conditions under which the tangent bundle

T M

has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle [...] Read more.

The purpose of this paper is to find some conditions under which the tangent bundle

T M

has a dualistic structure. Then, we introduce infinitesimal affine transformations on statistical manifolds and investigate these structures on a special statistical distribution and the tangent bundle of a statistical manifold too. Moreover, we also study the mutual curvatures of a statistical manifold M and its tangent bundle

T M

and we investigate their relations. More precisely, we obtain the mutual curvatures of well-known connections on the tangent bundle

T M

(the complete, horizontal, and Sasaki connections) and we study the vanishing of them. Full article

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

16 pages, 545 KiB

Open AccessArticle

Bertrand Offsets of Ruled Surfaces with Blaschke Frame in Euclidean 3-Space

by Sahar H. Nazra and Rashad A. Abdel-Baky

Axioms 2023, 12(7), 649; https://doi.org/10.3390/axioms12070649 - 29 Jun 2023

Cited by 3 | Viewed by 909

Abstract

Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled [...] Read more.

Dual representations of the Bertrand offset-surfaces are specified and several new results are gained in terms of their integral invariants. A new description of Bertrand offsets of developable surfaces is given. Furthermore, several relationships through the striction curves of Bertrand offsets of ruled surfaces and their integral invariants are obtained. Full article

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

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