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Symmetry/Asymmetry: Differential Geometry and Its Applications

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 19854

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Special Issue Editors

Prof. Dr. Luca Grilli

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Guest Editor

Department of Economy Management and Territory, University of Foggia, 71122 Foggia, Italy
Interests: game theory and applications; artificial intelligence; machine/deep learning; sustainability; finance; multicriteria decision; making; health economics
Special Issues, Collections and Topics in MDPI journals

Dr. Süleyman Şenyurt

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Guest Editor

Department of Mathematics, Ordu University, Ordu 52200, Turkey
Interests: differential geometry; Lorentz geometry

Prof. Dr. Marian Ioan Munteanu

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Guest Editor

Faculty of Mathematics, Alexandru Ioan Cuza University of Iasi, Bd. Carol I, N. 11, 700506 Iasi, Romania
Interests: geometry and topology; differential geometry; submanifolds
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential geometry is the branch of mathematics that studies the geometry of curves, surfaces, and manifolds (high-dimensional analogues of surfaces). Although the modern subject often uses algebraic and purely geometric techniques, the discipline owes its name to the use of ideas and techniques in differential calculus. The founder of differential geometry is considered to be Carl Friedrich Gauss. Gauss made important contributions to the field of differential geometry of curves and surfaces, and his work formed the basis of modern differential geometry. Differential geometry finds applications throughout mathematics and the natural sciences. Most prominently, the language of differential geometry was used by Albert Einstein in his theory of general relativity, and subsequently by physicists in the development of quantum field theory and the standard model of particle physics. Outside of physics, differential geometry finds applications in chemistry, economics, engineering, control theory, computer graphics and computer vision, and recently in machine learning. The use of differential geometry tools in computer science has become increasingly common in recent years, and the number of global investigations has grown. In addition to curves and surfaces, there are applications of manifolds theory in many fields, such as data analysis, image and audio processing, and data mining.

This Special Issue is devoted to discussing the latest technological developments as well as providing the latest findings related to various fields of differential geometry. Therefore, we would like to invite researchers from all over the world—especially those who use the concepts of symmetry or asymmetry in their methodologies—to share their work in this issue.

Dr. Luca Grilli
Dr. Süleyman Şenyurt
Prof. Dr. Marian Ioan Munteanu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

symmetric and a-symmetric curve/surface pairs
Lorentz-Minkowski space
timelike (spacelike) surfaces
timelike (spacelike) curves
dual space
invariants
singularities
spherical movements
E. Study map
congruances
ruled surfaces
striction curve
Blaschke frame
the theory of relativity
connetions
geodesic curvature
geodesics
metrics
parallel transport
lie derivative
geometric group/measure theory
manifolds and sub-manifolds
tensor analysis
Darboux frame
Enneper theorem
Euler-Savary formula
O. Bonnet theorem
Liouville theorem
Gaussian curvature
mean curvature
Frenet frame
modified frame
indicatrix curve
Bezier curves
computer graphics

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Published Papers (13 papers)

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Research

12 pages, 1920 KiB

Open AccessArticle

Timelike Surface Couple with Bertrand Couple as Joint Geodesic Curves in Minkowski 3-Space

by Fatemah Mofarreh

Symmetry 2024, 16(6), 732; https://doi.org/10.3390/sym16060732 - 12 Jun 2024

Viewed by 584

Abstract

A curve on a surface is a geodesic curve if its principal normal vector is anywhere aligned with the surface normal. Using the Serret–Frenet frame, a timelike surface couple (

TLSC)

with the symmetry of a Bertrand couple (

BC

) can [...] Read more.

A curve on a surface is a geodesic curve if its principal normal vector is anywhere aligned with the surface normal. Using the Serret–Frenet frame, a timelike surface couple (

TLSC)

with the symmetry of a Bertrand couple (

BC

) can be specified in terms of linear combinations of the components of the local frames in Minkowski 3-space

E_{1}^{3}

. With these parametric representations, the necessary and sufficient conditions for the specified

BC

are derived to be the geodesic curves defining these surfaces. Afterward, the definition of a

TL

ruled surface (

RS

) is also provided. Furthermore, the application of the method to some significant models is given. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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Figure 1

13 pages, 326 KiB

Open AccessArticle

Curve-Surface Pairs on Embedded Surfaces and Involute D-Scroll of the Curve-Surface Pair in E³

by Filiz Ertem Kaya and Süleyman Şenyurt

Symmetry 2024, 16(3), 323; https://doi.org/10.3390/sym16030323 - 7 Mar 2024

Cited by 1 | Viewed by 1024

Abstract

Willmore defined embedded surfaces on

f : S \to E^{3}

, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of

E^{3}

, inducing a Riemannian structure on

f (S)

. The expression analogous [...] Read more.

Willmore defined embedded surfaces on

f : S \to E^{3}

, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of

E^{3}

, inducing a Riemannian structure on

f (S)

. The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature

H^{2}

f (S) .

Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the

\tilde{D}

-scroll of the curve–surface pair and involute

\tilde{D}

-scroll of the curve–surface pair with some differential geometric elements and found

{\tilde{D}}_{(α, M)} (s)

and

{\tilde{D^{*}}}_{(α, M)} (s)

-scrolls of the curve–surface pair

(α, M)

. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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17 pages, 1290 KiB

Open AccessArticle

A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

by Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña and Beatriz Rubio

Symmetry 2023, 15(11), 2041; https://doi.org/10.3390/sym15112041 - 10 Nov 2023

Viewed by 961

Abstract

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space [...] Read more.

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space

U_{4} (I_{α}) = span {1, cos t, sin t, cos 2 t, sin 2 t}

defined on compact intervals

I_{α} = [0, α]

, where

α

is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space

U_{4} (I_{α})

to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as

α \to 0

. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of

C^{(3}

Hermite interpolation problems. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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16 pages, 295 KiB

Open AccessArticle

Proposed Theorems on the Lifts of Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection (NSNMC) in the Tangent Bundle

by Rajesh Kumar, Lalnunenga Colney and Mohammad Nazrul Islam Khan

Symmetry 2023, 15(11), 2037; https://doi.org/10.3390/sym15112037 - 9 Nov 2023

Cited by 5 | Viewed by 1340

Abstract

The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

17 pages, 339 KiB

Open AccessArticle

Three-Dimensional Semi-Symmetric Almost α-Cosymplectic Manifolds

by Sermin Öztürk and Hakan Öztürk

Symmetry 2023, 15(11), 2022; https://doi.org/10.3390/sym15112022 - 5 Nov 2023

Viewed by 896

Abstract

The main objective of this paper is to study semi-symmetric almost

α

-cosymplectic three-manifolds. We present basic formulas for almost

α

-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost

α

-cosymplectic three-manifolds. We obtain the main [...] Read more.

The main objective of this paper is to study semi-symmetric almost

α

-cosymplectic three-manifolds. We present basic formulas for almost

α

-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost

α

-cosymplectic three-manifolds. We obtain the main results under an additional condition. The paper concludes with two illustrative examples. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

13 pages, 2792 KiB

Open AccessArticle

The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space

by Serkan Çelik, Hacı Bayram Karadağ and Hatice Kuşak Samancı

Symmetry 2023, 15(6), 1191; https://doi.org/10.3390/sym15061191 - 2 Jun 2023

Cited by 2 | Viewed by 1373

Abstract

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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13 pages, 458 KiB

Open AccessArticle

On Disks Enclosed by Smooth Jordan Curves

by José Ayala Hoffmann and Victor Ayala

Symmetry 2023, 15(6), 1140; https://doi.org/10.3390/sym15061140 - 24 May 2023

Viewed by 1026

Abstract

Given a smooth-plane Jordan curve with bounded absolute curvature

κ > 0

, we determine equivalence classes of distinctive disks of radius

1 / κ

included in both plane regions separated by the curve. The bound on absolute curvature leads to a completely [...] Read more.

Given a smooth-plane Jordan curve with bounded absolute curvature

κ > 0

, we determine equivalence classes of distinctive disks of radius

1 / κ

included in both plane regions separated by the curve. The bound on absolute curvature leads to a completely symmetric trajectory behaviour with respect to the curve turning. These lead to a decomposition of the plane into a finite number of maximal regions with respect to set inclusion leading to natural lower bounds for the length an area enclosed by the curve. We present a “half version” of the Pestov–Ionin theorem, and subsequently a generalisation of the classical Blaschke rolling disk theorem. An interesting consequence is that we describe geometric conditions relying exclusively on curvature and independent of any kind of convexity that allows us to give necessary and sufficient conditions for the existence of families of rolling disks for planar domains that are not necessarily convex. We expect this approach would lead to further generalisations as, for example, characterising volumetric objects in closed surfaces as first studied by Lagunov. Although this is a classical problem in differential geometry, recent developments in industrial manufacturing when cutting along some prescribed shapes on prescribed materials have revived the necessity of a deeper understanding on disks enclosed by sufficiently smooth Jordan curves. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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12 pages, 15985 KiB

Open AccessArticle

Surfaces with Constant Negative Curvature

by Semra Kaya Nurkan and İbrahim Gürgil

Symmetry 2023, 15(5), 997; https://doi.org/10.3390/sym15050997 - 28 Apr 2023

Viewed by 1616

Abstract

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic

I I

-flat, isotropic minimal and isotropic

I I

-minimal, the constant second Gaussian curvature, [...] Read more.

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic

I I

-flat, isotropic minimal and isotropic

I I

-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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14 pages, 328 KiB

Open AccessArticle

Ricci Soliton of

CR

-Warped Product Manifolds and Their Classifications

by Yanlin Li, Sachin Kumar Srivastava, Fatemah Mofarreh, Anuj Kumar and Akram Ali

Symmetry 2023, 15(5), 976; https://doi.org/10.3390/sym15050976 - 25 Apr 2023

Cited by 28 | Viewed by 1860

Abstract

In this article, we derived an equality for

CR

-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a

CR

-warped product [...] Read more.

In this article, we derived an equality for

CR

-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a

CR

-warped product submanifolds in K

\ddot{a}

hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of

CR

-warped product submanifolds in the K

\ddot{a}

hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

18 pages, 773 KiB

Open AccessArticle

Algebraic Schouten Solitons of Three-Dimensional Lorentzian Lie Groups

by Siyao Liu

Symmetry 2023, 15(4), 866; https://doi.org/10.3390/sym15040866 - 5 Apr 2023

Viewed by 1002

Abstract

In 2016, Wears defined and studied algebraic T-solitons. In this paper, we define algebraic Schouten solitons as a special T-soliton and classify the algebraic Schouten solitons associated with Levi-Civita connections, canonical connections, and Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups that have some [...] Read more.

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

17 pages, 834 KiB

Open AccessArticle

Differentiating the State Evaluation Map from Matrices to Functions on Projective Space

by Ghaliah Alhamzi and Edwin Beggs

Symmetry 2023, 15(2), 474; https://doi.org/10.3390/sym15020474 - 10 Feb 2023

Cited by 1 | Viewed by 1042

Abstract

The pure state evaluation map from

M_{n} (C)

C (C P^{n - 1})

is a completely positive map of

C^{*}

-algebras intertwining the

U_{n}

symmetries on the two algebras. We show that it extends [...] Read more.

The pure state evaluation map from

M_{n} (C)

C (C P^{n - 1})

is a completely positive map of

C^{*}

-algebras intertwining the

U_{n}

symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on

M_{n} (C)

to the holomorphic

\bar{\partial}

calculus on

C P^{n - 1}

. The method uses connections on Hilbert

C^{*}

-bimodules. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

22 pages, 709 KiB

Open AccessArticle

The Invariants of Dual Parallel Equidistant Ruled Surfaces

by Sümeyye Gür Mazlum, Süleyman Şenyurt and Luca Grilli

Symmetry 2023, 15(1), 206; https://doi.org/10.3390/sym15010206 - 10 Jan 2023

Cited by 13 | Viewed by 1435

Abstract

In this paper, we calculate the Gaussian curvatures of the dual spherical indicatrix curves formed on unit dual sphere by the Blaschke vectors and dual instantaneous Pfaff vectors of dual parallel equidistant ruled surfaces (DPERS) and we give the relationships between these curvatures. In addition to—in cases where the base curves of these DPERS are closed—computing the dual integral invariants of the indicatrix curves. Additionally, we show the relationships between them. Finally, we provide an example for each of these indicatrix curves. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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33 pages, 2064 KiB

Open AccessArticle

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

by Elena Campo-Montalvo, Marián Fernández de Sevilla, J. Rafael Magdalena Benedicto and Sonia Pérez-Díaz

Symmetry 2023, 15(1), 69; https://doi.org/10.3390/sym15010069 - 26 Dec 2022

Cited by 2 | Viewed by 1821

Abstract

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve,

C

, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied [...] Read more.

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve,

C

, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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