Symmetry/Asymmetry: Differential Geometry and Its Applications

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 11461

Special Issue Editors

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

Published Papers (11 papers)

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Research

17 pages, 1290 KiB  
Article
A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves
Symmetry 2023, 15(11), 2041; https://doi.org/10.3390/sym15112041 - 10 Nov 2023
Viewed by 546
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 U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[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 U4(Iα) 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 α0. 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  
Article
Proposed Theorems on the Lifts of Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection (NSNMC) in the Tangent Bundle
Symmetry 2023, 15(11), 2037; https://doi.org/10.3390/sym15112037 - 09 Nov 2023
Cited by 2 | Viewed by 901
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 [...] Read more.
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  
Article
Three-Dimensional Semi-Symmetric Almost α-Cosymplectic Manifolds
Symmetry 2023, 15(11), 2022; https://doi.org/10.3390/sym15112022 - 05 Nov 2023
Viewed by 562
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  
Article
The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space
Symmetry 2023, 15(6), 1191; https://doi.org/10.3390/sym15061191 - 02 Jun 2023
Cited by 1 | Viewed by 871
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 [...] Read more.
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  
Article
On Disks Enclosed by Smooth Jordan Curves
Symmetry 2023, 15(6), 1140; https://doi.org/10.3390/sym15061140 - 24 May 2023
Viewed by 593
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  
Article
Surfaces with Constant Negative Curvature
Symmetry 2023, 15(5), 997; https://doi.org/10.3390/sym15050997 - 28 Apr 2023
Viewed by 986
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 II-flat, isotropic minimal and isotropic II-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 II-flat, isotropic minimal and isotropic II-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  
Article
Ricci Soliton of CR-Warped Product Manifolds and Their Classifications
Symmetry 2023, 15(5), 976; https://doi.org/10.3390/sym15050976 - 25 Apr 2023
Cited by 26 | Viewed by 1227
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 Ka¨hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka¨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  
Article
Algebraic Schouten Solitons of Three-Dimensional Lorentzian Lie Groups
Symmetry 2023, 15(4), 866; https://doi.org/10.3390/sym15040866 - 05 Apr 2023
Viewed by 668
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.
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 product structure. Full article
(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)
17 pages, 834 KiB  
Article
Differentiating the State Evaluation Map from Matrices to Functions on Projective Space
Symmetry 2023, 15(2), 474; https://doi.org/10.3390/sym15020474 - 10 Feb 2023
Cited by 1 | Viewed by 624
Abstract
The pure state evaluation map from Mn(C) to C(CPn1) is a completely positive map of C*-algebras intertwining the Un symmetries on the two algebras. We show that it extends [...] Read more.
The pure state evaluation map from Mn(C) to C(CPn1) is a completely positive map of C*-algebras intertwining the Un symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on Mn(C) to the holomorphic ¯ calculus on CPn1. 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  
Article
The Invariants of Dual Parallel Equidistant Ruled Surfaces
Symmetry 2023, 15(1), 206; https://doi.org/10.3390/sym15010206 - 10 Jan 2023
Cited by 11 | Viewed by 978
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. [...] Read more.
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  
Article
Some New Symbolic Algorithms for the Computation of Generalized Asymptotes
Symmetry 2023, 15(1), 69; https://doi.org/10.3390/sym15010069 - 26 Dec 2022
Cited by 2 | Viewed by 1276
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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