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Axioms
Special Issues
Research in Differential Geometry and Singularity Theory
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Special Issue "Research in Differential Geometry and Singularity Theory"

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Geometry and Topology".

Deadline for manuscript submissions: 29 February 2024 | Viewed by 1874

Special Issue Editors

Prof. Dr. Zhigang Wang

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Guest Editor
School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China
Interests: singularity theory; differential geometry
Prof. Dr. Yanlin Li

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

E-Mail Website
Guest Editor
Departamento de Geometria y Topologia, Facultad de Ciencias, Universidad de Granada, E-18071 Granada, Spain
Interests: differential geometry; Riemannian geometry; real hypersurfaces in symmetric spaces
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The geometry of curves and surfaces is a subject that has fascinated many mathematicians and users of mathematics. In recent years, there has been a new approach to this classical subject, from the point of view of singularity theory. For example, robust geometric features on a surface in the Euclidean 3-space, some of which are detectable by the naked eye, can be captured by certain types of singularities for some functions and mappings on the surface. The mappings in question are in fact members of some natural families of mappings on the surface. The singularities of the individual members of these families of mappings measure the contact of the surface with model objects, such as lines, circles, planes and spheres. It is interesting to investigate how to apply singularity theory to the study of the extrinsic geometry of surfaces, and how such methods are applied to any smooth submanifolds of higher-dimensional Euclidean space, as well as to other settings, such as affine, hyperbolic or Minkowski spaces. Singularities arise naturally in a huge number of different areas of mathematics and science. 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 fields of application, and are the breakthrough point in the discovery of 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. Currently, there is a growing and justified interest in the study of the differential geometry of singular submanifolds (such as caustics, wavefronts, images of singular mappings, etc.) of Euclidean or Minkowski spaces, and of submanifolds with induced (pseudo) metrics that change signature on some subsets of submanifolds. We hope this Special Issue can bring together experts within the field and those from adjacent areas where singularity theory has existing or potential applications. One of the aims of this Special Issue is to provide a platform for papers focused on differential geometry and singularity theory, devoted to surveying the remarkable insights in any related fields, and exploring promising new developments.

We look forward to receiving your contributions.

Prof. Dr. Zhigang Wang
Prof. Dr. Yanlin Li
Prof. Dr. Juan De Dios Pérez
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. Axioms 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 1600 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
  • singularities
  • singular submanifolds
  • lightlike submanifolds
  • differentiable manifolds
  • submanifold theory
  • legendrian duality
  • front and frontal

Published Papers (4 papers)

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Research

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Article
Confocal Families of Hyperbolic Conics via Quadratic Differentials
by
Joel Langer
and
David Singer
Axioms 2023, 12(6), 507; https://doi.org/10.3390/axioms12060507 - 24 May 2023
Viewed by 246
Abstract
We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that [...] Read more.
We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics. Full article
(This article belongs to the Special Issue Research in Differential Geometry and Singularity Theory)
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Article
Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces
by
Yanlin Li
and
Abdussamet Çalışkan
Axioms 2023, 12(5), 486; https://doi.org/10.3390/axioms12050486 - 17 May 2023
Cited by 1 | Viewed by 265
Abstract
In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, [...] Read more.
In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed. Full article
(This article belongs to the Special Issue Research in Differential Geometry and Singularity Theory)
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Article
On an Indefinite Metric on a Four-Dimensional Riemannian Manifold
by
Dimitar Razpopov
,
Georgi Dzhelepov
and
Iva Dokuzova
Axioms 2023, 12(5), 432; https://doi.org/10.3390/axioms12050432 - 27 Apr 2023
Viewed by 298
Abstract
Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type (1,1), whose fourth power is minus [...] Read more.
Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type (1,1), whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures define an indefinite metric. With the help of the indefinite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we define a vector force field. Further, we obtain the circulation of the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space. Full article
(This article belongs to the Special Issue Research in Differential Geometry and Singularity Theory)
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Article
C-R Immersions and Sub-Riemannian Geometry
by
Elisabetta Barletta
,
Sorin Dragomir
and
Francesco Esposito
Axioms 2023, 12(4), 329; https://doi.org/10.3390/axioms12040329 - 28 Mar 2023
Viewed by 549
Abstract
On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm [...] Read more.
On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form θ, we consider natural ϵ-contractions, i.e., contractions gϵM of the Levi form Gθ, such that the norm of the Reeb vector field T of (M, θ) is of order O(ϵ−1). We study isopseudohermitian (i.e., f∗Θ=θ) Cauchy–Riemann immersions f:M→(A,Θ) between strictly pseudoconvex CR manifolds M and A, where Θ is a contact form on A. For every contraction gϵA of the Levi form GΘ, we write the embedding equations for the immersion f:M→A,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ→0+. For every isopseudohermitian immersion f:M→S2N+1 into a sphere S2N+1⊂CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R≤2n(f∗gΘ)(T,T)+n+1+12{∥H(f)∥gΘf2+∥traceGθΠH(M)∇⊤−∇∥f∗gΘ2} with equality if and only if B(f)=0 and ∇⊤=∇ on H(M)⊗H(M). This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms. Full article
(This article belongs to the Special Issue Research in Differential Geometry and Singularity Theory)
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