Advances 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: 31 July 2024 | Viewed by 12770

Special Issue Editors


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Guest Editor
School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China
Interests: singularity theory; differential geometry

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Guest Editor
Departamento de Geometria y Topologia, Facultad de Ciencias, 18071 Granada, Spain
Interests: differential geometry; Riemannian geometry; real hypersurfaces i 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 However, due to the existence of singular sets, the traditional analysis and geometric mathematical tools are no longer applicable, which makes the study of singular submanifolds difficult.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

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Keywords

  • singularity theory
  • morse theory
  • singularities
  • singular submanifolds
  • lightlike submanifolds
  • differentiable manifolds
  • submanifold theory
  • legendrian duality
  • front and frontal

Published Papers (14 papers)

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Research

13 pages, 261 KiB  
Article
On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities
by Naveed Hussain, Ahmad N. Al-Kenani and Muhammad Asif
Axioms 2024, 13(4), 216; https://doi.org/10.3390/axioms13040216 - 25 Mar 2024
Viewed by 459
Abstract
Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of classification theory of isolated singularities. Any Lie algebra that [...] Read more.
Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of classification theory of isolated singularities. Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work. Hussain, Yau and Zuo have found a new class of Lie algebra Lk(V), k1, i.e., Der (Mk(V),Mk(V)) and proposed a conjecture over its dimension δk(V) for k0. Later, they proved it true for k up to k=1,2,3,4,5. In this work, the main concern is whether it is true for a higher value of k. According to this, we first calculate the dimension of Lie algebra Lk(V) for k=6 and then compute the upper estimate conjecture of fewnomial isolated singularities. Additionally, we also justify the inequality conjecture δk+1(V)<δk(V) for k=6. Our calculated results are innovative and serve as a new addition to the study of singularity theory. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
11 pages, 243 KiB  
Article
The Bifurcations of Completely Integrable Holonomic Systems of First-Order Differential Equations
by Jingbo Xu, Kangping Liu and Xiaoliang Cheng
Axioms 2024, 13(1), 55; https://doi.org/10.3390/axioms13010055 - 16 Jan 2024
Viewed by 761
Abstract
As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are R+-simple and stable. Using this result, the parametric differential equation [...] Read more.
As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are R+-simple and stable. Using this result, the parametric differential equation models in electrical power systems and engineering can be studied. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
20 pages, 363 KiB  
Article
Optimal Inequalities on (α,β)-Type Almost Contact Manifold with the Schouten–Van Kampen Connection
by Mohd Danish Siddiqi and Ali H. Hakami
Axioms 2023, 12(12), 1082; https://doi.org/10.3390/axioms12121082 - 27 Nov 2023
Viewed by 1061
Abstract
In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or (α,β)-type almost contact manifolds endowed with the Schouten–Van Kampen connection (SVK-connection), including generalized normalized δ-Casorati Curvatures (δ-CC). [...] Read more.
In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or (α,β)-type almost contact manifolds endowed with the Schouten–Van Kampen connection (SVK-connection), including generalized normalized δ-Casorati Curvatures (δ-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
15 pages, 279 KiB  
Article
Semi-Conformally Flat Singly Warped Product Manifolds and Applications
by Samesh Shenawy, Alaa Rabie, Uday Chand De, Carlo Mantica and Nasser Bin Turki
Axioms 2023, 12(12), 1078; https://doi.org/10.3390/axioms12121078 - 24 Nov 2023
Viewed by 807
Abstract
This paper investigates singly warped product manifolds admitting semi-conformal curvature tensors. The form of the Riemann tensor and Ricci tensor of the base and fiber manifolds of a semi-conformally flat singly warped product manifold are provided. It is demonstrated that the fiber manifold [...] Read more.
This paper investigates singly warped product manifolds admitting semi-conformal curvature tensors. The form of the Riemann tensor and Ricci tensor of the base and fiber manifolds of a semi-conformally flat singly warped product manifold are provided. It is demonstrated that the fiber manifold of a semi-conformally flat warped product manifold has a constant curvature. Sufficient requirements on the warping function to ensure that the base manifold is a quasi-Einstein or an Einstein manifold are provided. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
16 pages, 621 KiB  
Article
On the Timelike Circular Surface and Singularities in Minkowski 3-Space
by Areej A. Almoneef and Rashad A. Abdel-Baky
Axioms 2023, 12(10), 989; https://doi.org/10.3390/axioms12100989 - 19 Oct 2023
Viewed by 824
Abstract
In this paper, we have parameterized a timelike (Tlike) circular surface (CIsurface) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a [...] Read more.
In this paper, we have parameterized a timelike (Tlike) circular surface (CIsurface) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a Tlike roller coaster surface (RCOsurface) to be a flat or minimal surface is examined in detail. Further, we illustrate the approach’s outcomes with a number of pertinent examples. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
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14 pages, 287 KiB  
Article
Inequalities for the Generalized Normalized δ-Casorati Curvatures of Submanifolds in Golden Riemannian Manifolds
by Majid Ali Choudhary and Ion Mihai
Axioms 2023, 12(10), 952; https://doi.org/10.3390/axioms12100952 - 08 Oct 2023
Viewed by 723
Abstract
In the present article, we consider submanifolds in golden Riemannian manifolds with constant golden sectional curvature. On such submanifolds, we prove geometric inequalities for the Casorati curvatures. The submanifolds meeting the equality cases are also described. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
55 pages, 531 KiB  
Article
On the Geometry of the Null Tangent Bundle of a Pseudo-Riemannian Manifold
by Mohamed Tahar Kadaoui Abbassi, Khadija Boulagouaz and Giovanni Calvaruso
Axioms 2023, 12(10), 903; https://doi.org/10.3390/axioms12100903 - 22 Sep 2023
Viewed by 534
Abstract
When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on [...] Read more.
When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on the tangent bundle of a pseudo-Riemannian manifold and we investigate the geometry of the null tangent bundle as a lightlike hypersurface equipped with an induced g-natural metric. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
10 pages, 926 KiB  
Article
Surface Family Pair with Bertrand Pair as Mutual Curvature Lines in Three-Dimensional Lie Group
by Awatif Al-Jedani and Rashad A. Abdel-Baky
Axioms 2023, 12(9), 830; https://doi.org/10.3390/axioms12090830 - 28 Aug 2023
Viewed by 531
Abstract
This paper is on deducing the necessary and sufficient conditions of a surface family pair with a Bertrand pair as mutual curvature lines in three-dimensional Lie group G. As a result, the consequence for the ruled surface family pair is also extrapolated. [...] Read more.
This paper is on deducing the necessary and sufficient conditions of a surface family pair with a Bertrand pair as mutual curvature lines in three-dimensional Lie group G. As a result, the consequence for the ruled surface family pair is also extrapolated. Meanwhile, examples are specified to show the surface family with common Bertrand geodesic curves. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
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15 pages, 280 KiB  
Article
On Some Quasi-Curves in Galilean Three-Space
by Ayman Elsharkawy, Yusra Tashkandy, Walid Emam, Clemente Cesarano and Noha Elsharkawy
Axioms 2023, 12(9), 823; https://doi.org/10.3390/axioms12090823 - 27 Aug 2023
Viewed by 887
Abstract
In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute [...] Read more.
In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated example to present our findings. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
11 pages, 272 KiB  
Article
Kropina Metrics with Isotropic Scalar Curvature
by Liulin Liu, Xiaoling Zhang and Lili Zhao
Axioms 2023, 12(7), 611; https://doi.org/10.3390/axioms12070611 - 21 Jun 2023
Cited by 1 | Viewed by 758
Abstract
In this paper, we study Kropina metrics with isotropic scalar curvature. First, we obtain the expressions of Ricci curvature tensor and scalar curvature. Then, we characterize the Kropina metrics with isotropic scalar curvature on by tensor analysis. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory)
14 pages, 3785 KiB  
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
Cited by 1 | Viewed by 941
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 Advances in Differential Geometry and Singularity Theory)
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16 pages, 403 KiB  
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 18 | Viewed by 1122
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 Advances in Differential Geometry and Singularity Theory)
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13 pages, 305 KiB  
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 723
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 Advances in Differential Geometry and Singularity Theory)
82 pages, 748 KiB  
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 962
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(ϵ1). We study isopseudohermitian (i.e., fΘ=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:MA,gϵA. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as ϵ0+. For every isopseudohermitian immersion f:MS2N+1 into a sphere S2N+1CN+1, we show that Webster’s pseudohermitian scalar curvature R of (M, θ) satisfies the inequality R2n(fgΘ)(T,T)+n+1+12{H(f)gΘf2+traceGθΠH(M)fgΘ2} 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 Advances in Differential Geometry and Singularity Theory)
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