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Symmetry
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Symmetry and Its Application in Differential Geometry and Topology
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Symmetry and Its Application in Differential Geometry and Topology

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

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 31207

Special Issue Editors

Dr. Yanlin Li

E-Mail Website
Guest Editor
Associate Professor, School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
Interests: singularity theory; differential geometry
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Tiehong Zhao

E-Mail Website
Guest Editor
School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
Interests: differential geometry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential geometry is a branch of mathematics that has many applications not only in mathematics but in many other sciences, e.g., applications of the theory of curves and surfaces in the Euclidean plane and space. Geometry and Topology are quite related to Symmetry. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis. Differential geometry can be defined as the study of the geometry of differential manifolds, as well as of their submanifolds. In recent years, there has been a fast-growing interest in developing theories and tools for studying singular submanifolds. Because singular submanifolds are produced in physics, mechanics, and other application fields and are the breakthrough point to discover new problems. Therefore, it is of great scientific significance to study the geometric and topological properties of singular submanifolds. However, due to the existence of singular sets, the traditional analysis and geometric mathematical tools are no longer applicable, which makes the study of singular submanifolds difficult. In addition, applications of differential geometry and Topology can be found in almost any field of science, from biology to architecture. One of the most important applications of Topology is Topological Data Analysis (TDA). TDA combines ideas from Topology and also algebra, geometry, and analysis, with methods from statistics and computer science, for the purpose of analyzing contemporary data sets for which standard approaches are unsatisfactory. The motivating idea is that there is an underlying ''shape'' to the data and that new variants of some of the sophisticated tools of modern mathematics may be brought to bear to elucidate and learn from this structure. TDA has convincingly proved its utility in a wide range of applications in the life sciences, including in neuroscience, genomics, proteomics, evolution, and cancer biology, among other areas of research.

This Special Issue is intended to provide a series of papers focused on Symmetry and its applications of geometry and Topology, devoted to surveying the remarkable insights into many fields of sciences and exploring promising new developments.

Dr. Yanlin Li
Prof. Dr. Tiehong Zhao
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

  • Singularity theory
  • Morse theory/ Discrete Morse theory
  • Singularities
  • Singular submanifolds
  • Lightlike submanifolds
  • Biharmonic submanifolds
  • Warped product submanifolds
  • Differentiable manifolds
  • Submanifold theory
  • Legendrian duality
  • Front and frontal
  • Physics
  • Statistics
  • Topological data analysis
  • Computational topology
  • Applied topology and geometry
  • Topological and geometric methods in data analysis
  • Spectral and geometric methods in machine learning and data analysis
  • Persistent homology and cohomology, and applications
  • Neuroscience
  • Cancer biology
  • Genomics
  • Other sciences

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Published Papers (17 papers)

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Research

7 pages, 306 KiB  
Article
On the Computation of the Codimension of Map Germs Using the Lie Algebra Associated with a Restricted Left–Right Group
by Peng Xu, Muhammad Ahsan Binyamin and Adnan Aslam
Symmetry 2023, 15(5), 1042; https://doi.org/10.3390/sym15051042 - 8 May 2023
Cited by 1 | Viewed by 1238
Abstract
The codimension is an important invariant, which measures the complexity of map germs and play an important role in classification and recognition problems. The restricted A-equivalence was introduced to obtain a classification of reducible curves. The aim was to classify simple parameterized [...] Read more.
The codimension is an important invariant, which measures the complexity of map germs and play an important role in classification and recognition problems. The restricted A-equivalence was introduced to obtain a classification of reducible curves. The aim was to classify simple parameterized curves with two components, one of them being smooth with respect to the A-equivalence in characteristic p. In characteristic 0, the corresponding classification was given by Kolgushkin and Sadykov. The aim of this article is to present an algorithm to compute the codimension of germs of singularities under a restricted left–right equivalence (A-symmetry). We also give the implementation of this algorithm in the computer algebra system singular. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
13 pages, 669 KiB  
Article
Singularities for Timelike Developable Surfaces in Minkowski 3-Space
by Yanlin Li, Zhizhi Chen, Sahar H. Nazra and Rashad A. Abdel-Baky
Symmetry 2023, 15(2), 277; https://doi.org/10.3390/sym15020277 - 19 Jan 2023
Cited by 38 | Viewed by 1799
Abstract
In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is [...] Read more.
In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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15 pages, 297 KiB  
Article
Lightlike Hypersurfaces of Almost Productlike Semi-Riemannian Manifolds
by Ömer Aksu, Mehmet Gülbahar and Esra Erkan
Symmetry 2023, 15(1), 77; https://doi.org/10.3390/sym15010077 - 27 Dec 2022
Cited by 2 | Viewed by 1382
Abstract
The main purpose of this paper is to investigate lightlike hypersurfaces of almost productlike semi-Riemannian manifolds. For this purpose, screen-semi-invariant, screen-invariant, radical-anti-invariant, and radical-invariant lightlike hypersurfaces of almost productlike semi-Riemannian manifolds are introduced and some examples of these classifications are presented. Furthermore, various [...] Read more.
The main purpose of this paper is to investigate lightlike hypersurfaces of almost productlike semi-Riemannian manifolds. For this purpose, screen-semi-invariant, screen-invariant, radical-anti-invariant, and radical-invariant lightlike hypersurfaces of almost productlike semi-Riemannian manifolds are introduced and some examples of these classifications are presented. Furthermore, various characterizations dealing screen semi-invariant lightlike hypersurfaces are obtained. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
16 pages, 419 KiB  
Article
Singularities for Focal Sets of Timelike Sabban Curves in de Sitter 3-Space
by Yongqiao Wang, Lin Yang, Yuxin Liu and Yuan Chang
Symmetry 2022, 14(12), 2471; https://doi.org/10.3390/sym14122471 - 22 Nov 2022
Cited by 3 | Viewed by 1177
Abstract
In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and [...] Read more.
In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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9 pages, 250 KiB  
Article
Double Contingency of Communications in Bayesian Learning
by Atsuhide Mori
Symmetry 2022, 14(11), 2456; https://doi.org/10.3390/sym14112456 - 19 Nov 2022
Viewed by 1598
Abstract
In previous work, we described the geometry of Bayesian learning on a manifold. In this paper, inspired by the notion of modified double contingency of communications from sociologist Niklas Luhmann, we take two manifolds in equal parts and a potential function on their [...] Read more.
In previous work, we described the geometry of Bayesian learning on a manifold. In this paper, inspired by the notion of modified double contingency of communications from sociologist Niklas Luhmann, we take two manifolds in equal parts and a potential function on their product to set up mutual Bayesian learning. Particularly, given a parametric statistical model, we consider mutual learning between two copies of the parameter space. Here, we associate the potential with the relative entropy (i.e., the Kullback–Leibler divergence). Although the mutual learning forgets all elements about the model except the relative entropy, it still substitutes for the usual Bayesian estimation of the parameter in a certain case. We propose it as a globalization of the information geometry. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
21 pages, 546 KiB  
Article
Singularities of Osculating Developable Surfaces of Timelike Surfaces along Curves
by Yongqiao Wang, Lin Yang, Pengcheng Li and Yuan Chang
Symmetry 2022, 14(11), 2251; https://doi.org/10.3390/sym14112251 - 26 Oct 2022
Cited by 3 | Viewed by 1375
Abstract
In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel [...] Read more.
In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the osculating Darboux vector field. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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13 pages, 259 KiB  
Article
Classification of Lorentzian Lie Groups Based on Codazzi Tensors Associated with Yano Connections
by Jiajing Miao, Jinli Yang and Jianyun Guan
Symmetry 2022, 14(8), 1730; https://doi.org/10.3390/sym14081730 - 18 Aug 2022
Cited by 1 | Viewed by 1582
Abstract
In this paper, we derive the expressions of Codazzi tensors associated with Yano connections in seven Lorentzian Lie groups. Furthermore, we complete the classification of three-dimensional Lorentzian Lie groups in which Ricci tensors associated with Yano connections are Codazzi tensors. The main results [...] Read more.
In this paper, we derive the expressions of Codazzi tensors associated with Yano connections in seven Lorentzian Lie groups. Furthermore, we complete the classification of three-dimensional Lorentzian Lie groups in which Ricci tensors associated with Yano connections are Codazzi tensors. The main results are listed in a table, and indicate that G1 and G7 do not have Codazzi tensors associated with Yano connections, G2, G3, G4, G5 and G6 have Codazzi tensors associated with Yano connections. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
12 pages, 273 KiB  
Article
V-Quasi-Bi-Slant Riemannian Maps
by Sushil Kumar, Mohd Bilal, Rajendra Prasad, Abdul Haseeb and Zhizhi Chen
Symmetry 2022, 14(7), 1360; https://doi.org/10.3390/sym14071360 - 1 Jul 2022
Cited by 3 | Viewed by 1467
Abstract
In this work, we define a v-quasi-bi-slant Riemannian map (in brief, v-QBSR map) from almost Hermitian manifolds to Riemannian manifolds. This notion generalizes both a v-hemi slant Riemannian map and a v-semi slant Riemannian map. The geometry of leaves of distributions that are [...] Read more.
In this work, we define a v-quasi-bi-slant Riemannian map (in brief, v-QBSR map) from almost Hermitian manifolds to Riemannian manifolds. This notion generalizes both a v-hemi slant Riemannian map and a v-semi slant Riemannian map. The geometry of leaves of distributions that are associated with the definition of such maps is studied. The conditions for v-QBSR maps to be integrable and totally geodesic are also obtained in the paper. Finally, we provide the examples of v-QBSR maps. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
17 pages, 1319 KiB  
Article
Singularities of Slant Focal Surfaces along Lightlike Locus on Mixed Type Surfaces
by Haibo Yu and Liang Chen
Symmetry 2022, 14(6), 1203; https://doi.org/10.3390/sym14061203 - 10 Jun 2022
Viewed by 1517
Abstract
There are generally the mixed type surfaces with lightlike locus in the Lorentz-Minkowski 3-space. To investigate the geometry of lightlike locus, we define slant focal surfaces and slant evolutes associated to the oringinal mixed type surface by using a moving frame field along [...] Read more.
There are generally the mixed type surfaces with lightlike locus in the Lorentz-Minkowski 3-space. To investigate the geometry of lightlike locus, we define slant focal surfaces and slant evolutes associated to the oringinal mixed type surface by using a moving frame field along the lightlike locus defined by Honda etc. We obtain that singularities of slant focal surfaces and slant evolutes depend on the differential geometric properties of the lightlike locus. Furthermore, we investigate the relationship between slant focal surfaces and slant evolutes. We also consider the relationship between slant evolutes and the lightlike locus on the lightcone. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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24 pages, 1362 KiB  
Article
The Dual Expression of Parallel Equidistant Ruled Surfaces in Euclidean 3-Space
by Sümeyye Gür Mazlum, Süleyman Şenyurt and Luca Grilli
Symmetry 2022, 14(5), 1062; https://doi.org/10.3390/sym14051062 - 22 May 2022
Cited by 28 | Viewed by 2578
Abstract
In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere [...] Read more.
In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere corresponds to the p-distance. We call these ruled surfaces we obtained “dual parallel equidistant ruled surfaces” and we briefly denote them with “DPERS”. Furthermore, we find the Blaschke vectors, the Blaschke invariants and the striction curves of these DPERS and we give the relationships between these elements. Moreover, we show the relationships between the Darboux screws, the instantaneous screw axes, the instantaneous dual Pfaff vectors and dual Steiner rotation vectors of these surfaces. Finally, we give an example, which we reinforce this article, and we explain all of these features with the figures on the example. Furthermore, we see that the corresponding dual curves on the dual unit sphere to these DPERS are such that one of them is symmetric with respect to the imaginary symmetry axis of the other. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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21 pages, 386 KiB  
Article
Characterizations of PR-Pseudo-Slant Warped Product Submanifold of Para-Kenmotsu Manifold with Slant Base
by Sachin Kumar Srivastava, Fatemah Mofarreh, Anuj Kumar and Akram Ali
Symmetry 2022, 14(5), 1001; https://doi.org/10.3390/sym14051001 - 14 May 2022
Cited by 5 | Viewed by 1906
Abstract
In this article, we study the properties of PR-pseudo-slant submanifold of para-Kenmotsu manifold and obtain the integrability conditions for the slant distribution and anti-invariant distribution of such submanifold. We derived the necessary and sufficient conditions for a PR-pseudo-slant submanifold of para-Kenmotsu [...] Read more.
In this article, we study the properties of PR-pseudo-slant submanifold of para-Kenmotsu manifold and obtain the integrability conditions for the slant distribution and anti-invariant distribution of such submanifold. We derived the necessary and sufficient conditions for a PR-pseudo-slant submanifold of para-Kenmotsu manifold to be a PR-pseudo-slant warped product which are in terms of warping functions and shape operator. Some examples of PR-pseudo-slant warped products of para-Kenmotsu manifold are also illustrated in the article. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
14 pages, 312 KiB  
Article
Extended Legendrian Dualities Theorem in Singularity Theory
by Haiming Liu and Jiajing Miao
Symmetry 2022, 14(5), 982; https://doi.org/10.3390/sym14050982 - 11 May 2022
Cited by 1 | Viewed by 1367
Abstract
In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display [...] Read more.
In this paper, we find some new information on Legendrian dualities and extend them to the case of Legendrian dualities for continuous families of pseudo-spheres in general semi-Euclidean space. In particular, we construct all contact diffeomorphic mappings between the contact manifolds and display them in a table that contains all information about Legendrian dualities. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
14 pages, 405 KiB  
Article
Geometry of Developable Surfaces of Frenet Type Framed Base Curves from the Singularity Theory Viewpoint
by Qiming Zhao, Lin Yang and Yongqiao Wang
Symmetry 2022, 14(5), 975; https://doi.org/10.3390/sym14050975 - 10 May 2022
Cited by 2 | Viewed by 1534
Abstract
In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and [...] Read more.
In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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15 pages, 1266 KiB  
Article
A Study on the Bertrand Offsets of Timelike Ruled Surfaces in Minkowski 3-Space
by Rashad A. Abdel-Baky and Fatemah Mofarreh
Symmetry 2022, 14(4), 783; https://doi.org/10.3390/sym14040783 - 9 Apr 2022
Cited by 5 | Viewed by 1869
Abstract
This work extends some classical results of Bertrand curves to timelike ruled and developable surfaces using the E. Study map. This provides support to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is proved that every timelike [...] Read more.
This work extends some classical results of Bertrand curves to timelike ruled and developable surfaces using the E. Study map. This provides support to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is proved that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied among their dual invariants. In addition, some new results and theorems concerning the developability of the Bertrand offsets of timelike ruled surfaces are gained. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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14 pages, 1514 KiB  
Article
Kinematic Geometry of Timelike Ruled Surfaces in Minkowski 3-Space E13
by Nadia Alluhaibi and Rashad A. Abdel-Baky
Symmetry 2022, 14(4), 749; https://doi.org/10.3390/sym14040749 - 5 Apr 2022
Cited by 8 | Viewed by 2045
Abstract
Symmetry is a frequently recurring theme in mathematics, nature, science, etc. In mathematics, its most familiar manifestation appears in geometry, most notably line geometry, and in other closely related areas. In this study, we take advantage of the symmetry properties of both dual [...] Read more.
Symmetry is a frequently recurring theme in mathematics, nature, science, etc. In mathematics, its most familiar manifestation appears in geometry, most notably line geometry, and in other closely related areas. In this study, we take advantage of the symmetry properties of both dual space and original space in order to transfer problems in original space to dual space. We use E. Study Mappingas a direct method for analyzing the kinematic geometry of timelike ruled and developable surfaces. Then, the invariants for a spacelike line trajectory are studied and the well-known formulae of Hamilton and Mannheim on the theory of surfaces are provenfor the line space. Meanwhile, a timelike Plücker conoid generated by the Disteli-axis is derived and its kinematic geometry is discussed. Finally, some equations for particular timelike ruled surfaces, such as the general timelike helicoid, the Lorentzian sphere, and the timelike cone, are derived and plotted. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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11 pages, 468 KiB  
Article
Singularities of Non-Developable Ruled Surface with Space-like Ruling
by Rashad Abdel-Satar Abdel-Baky and Mohamed Khalifa Saad
Symmetry 2022, 14(4), 716; https://doi.org/10.3390/sym14040716 - 1 Apr 2022
Cited by 6 | Viewed by 2035
Abstract
Singularity theory is a significant field of modern mathematical research. The main goal in most problems of singularity theory is to understand the dependence of some objects in analysis and geometry, or physics; or from some other science on parameters. In this paper, [...] Read more.
Singularity theory is a significant field of modern mathematical research. The main goal in most problems of singularity theory is to understand the dependence of some objects in analysis and geometry, or physics; or from some other science on parameters. In this paper, we study the singularities of the spherical indicatrix and evolute of space-like ruled surface with space-like ruling. The main method takes advantage of the classical unfolding theorem in singularity theory, which is a classical method to study singularity problems in Euclidean space and Minkowski space. Finally, we provide an example to illustrate our results. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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16 pages, 1268 KiB  
Article
On the Bertrand Offsets of Timelike Ruled Surfaces in Minkowski 3-Space
by Nadia Alluhaibi, Rashad A. Abdel-Baky and Monia Naghi
Symmetry 2022, 14(4), 673; https://doi.org/10.3390/sym14040673 - 24 Mar 2022
Cited by 9 | Viewed by 1918
Abstract
This work examines some classical results of Bertrand curves for timelike ruled and developable surfaces using the E. Study map. This provides the ability to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is shown that every [...] Read more.
This work examines some classical results of Bertrand curves for timelike ruled and developable surfaces using the E. Study map. This provides the ability to define two timelike ruled surfaces which are offset in the sense of Bertrand. It is shown that every timelike ruled surface has a Bertrand offset if and only if an equation should be satisfied between the dual geodesic curvatures. Some new results and theorems related to the developability of the Bertrand offsets of timelike ruled surfaces are also obtained. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology)
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