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Symmetry
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Symmetry and Its Application in Differential Geometry and Topology, 4th...
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Symmetry and Its Application in Differential Geometry and Topology, 4th Edition

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

Deadline for manuscript submissions: 31 December 2025 | Viewed by 640

Special Issue Editors

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. 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 However, due to the existence of singular sets, the traditional analysis and geometric mathematical tools are no longer applicable, which makes the study of singular submanifolds difficult.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 (2 papers)

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Research

16 pages, 328 KiB  
Article
Null Hybrid Curves and Some Characterizations of Null Hybrid Bertrand Curves
by Jeta Alo
Symmetry 2025, 17(2), 312; https://doi.org/10.3390/sym17020312 - 19 Feb 2025
Viewed by 154
Abstract
In this paper, we investigate null curves in R24, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in R24. We [...] Read more.
In this paper, we investigate null curves in R24, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in R24. We first derive Frenet formulas for a null curve in R23, the three-dimensional Minkowski space of index 2, by means of spatial hybrid numbers. Next, we apply the Frenet formulas for the associated null spatial hybrid curve corresponding to a null hybrid curve in order to derive the Frenet formulas for this curve in R24. This approach is simpler and more efficient than the classical differential geometry methods and enables us to determine a null curve in R23 corresponding to the null curve in R24. Additionally, we provide an example of a null hybrid curve, demonstrate the construction of its Frenet frame, and calculate the curvatures of the curve. Finally, we introduce null hybrid Bertrand curves, and by using their symmetry properties, we provide some characterizations of these curves. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)
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14 pages, 276 KiB  
Article
Eigenvalues for the Generalized Laplace Operator of Slant Submanifolds in the Sasakian Space Forms Admitting Semi-Symmetric Metric Connection
by Ibrahim Al-Dayel, Meraj Ali Khan and Sudhakar Kumar Chaubey
Symmetry 2025, 17(2), 279; https://doi.org/10.3390/sym17020279 - 11 Feb 2025
Viewed by 347
Abstract
This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine [...] Read more.
This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine the principal eigenvalue for the generalized-Laplacian operator in closed oriented-slant submanifolds within a Sasakian space form (ssf) with a semi-symmetric metric (ssm) connection. By utilizing our findings on the Laplacian, we extend several Reilly-type inequalities to the generalized Laplacian on slant submanifolds within a unit sphere with a semi-symmetric metric (ssm) connection. The research is concluded with a detailed examination of specific scenarios. Full article
(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)
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