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Mathematics
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Geometric Topology and Differential Geometry with Applications
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Geometric Topology and Differential Geometry with Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "B: Geometry and Topology".

Deadline for manuscript submissions: 20 June 2025 | Viewed by 1562

Special Issue Editor

Dr. Daniele Ettore Otera

E-Mail Website
Guest Editor
Faculty of Mathematics and Informatics, Vilnius University, 03225 Vilnius, Lithuania
Interests: geometric group theory; low-dimensional topology; geometric topology; group theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleague,

Differential geometry is a branch of mathematics that makes use of differential and integral calculus to study and solve problems involving measurements in space.

The history of differential geometry goes hand in hand, at least in the seventeenth and eighteenth centuries, with advances in the fields of analytic geometry and differential calculus.

Indeed, differential geometry began as the study of curves and surfaces whose properties, varying from point to point, are investigated by using methods of differential calculus.

Therefore, it is in calculus that differential geometry has often found the origin of its problems and the methods for solving them.

Today, due to the power of analysis techniques, they have found applications in a variety of fields, and differential geometry is undoubtedly a central discipline in modern mathematics.

On the other hand, geometric topology is a more modern branch of geometry that studies the properties of manifolds that remain invariant under topological transformations.

Already implicit in the past, geometric topology was formalized with the pioneering work of Henri Poincaré in the late 1800s and then further developed by Felix Klein within the famous Erlangen Program in the early 1900s.

Today, after the spectacular advances made in the last century through the work of R. Thom, J. Milnor, S. Smale, M. Gromov, W. Thurston, S. Yau, S. Donaldson, E. Witten and G. Perelman, geometric topology has assumed a primary role in the modern landscape of mathematics and is closely related to the structure of manifolds of three and four dimensions.

The purpose of this Special Issue is to report some recent advances in these research fields, together with their applications.

Dr. Daniele Ettore Otera
Guest Editor

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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • manifolds
  • curvature
  • geometric analysis
  • fundamental group
  • Riemannian metrics
  • low-dimensional topology
  • geometric group theory

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Published Papers (3 papers)

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Research

8 pages, 239 KiB  
Article
Characterization of Ricci Solitons and Harmonic Vector Fields on the Lie Group Nil4
by Yanlin Li, Ahmed Mohammed Cherif and Yuquan Xie
Mathematics 2025, 13(7), 1155; https://doi.org/10.3390/math13071155 - 31 Mar 2025
Viewed by 201
Abstract
This study considers a left-invariant Riemannian metric g on the Lie group Nil4. We introduce a Ricci solitons’ classification on (Nil4,g). These are expansive non-gradient Ricci solitons. We examine the existence [...] Read more.
This study considers a left-invariant Riemannian metric g on the Lie group Nil4. We introduce a Ricci solitons’ classification on (Nil4,g). These are expansive non-gradient Ricci solitons. We examine the existence of harmonic maps into (Nil4,g) from a compact Riemannian manifold. Additionally, we provide a characterization of a class of harmonic vector fields on (Nil4,g). Full article
(This article belongs to the Special Issue Geometric Topology and Differential Geometry with Applications)
16 pages, 248 KiB  
Article
Conformal Interactions of Osculating Curves on Regular Surfaces in Euclidean 3-Space
by Yingxin Cheng, Yanlin Li, Pushpinder Badyal, Kuljeet Singh and Sandeep Sharma
Mathematics 2025, 13(5), 881; https://doi.org/10.3390/math13050881 - 6 Mar 2025
Cited by 3 | Viewed by 480
Abstract
Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and [...] Read more.
Conformal maps preserve angles and maintain the local shape of geometric structures. The osculating curve plays an important role in analyzing the variations in curvature, providing a detailed understanding of the local geometric properties and the impact of conformal transformations on curves and surfaces. In this paper, we study osculating curves on regular surfaces under conformal transformations. We obtained the conditions required for osculating curves on regular surfaces R and R˜ to remain invariant when subjected to a conformal transformation ψ:R→R˜. The results presented in this paper reveal the specific conditions under which the transformed curve σ˜=ψ∘σ preserves its osculating properties, depending on whether σ˜ is a geodesic, asymptotic, or neither. Furthermore, we analyze these conditions separately for cases with zero and non-zero normal curvatures. We also explore the behavior of these curves along the tangent vector Tσ and the unit normal vector Pσ. Full article
(This article belongs to the Special Issue Geometric Topology and Differential Geometry with Applications)
10 pages, 247 KiB  
Article
Dual Connectivity in Graphs
by Mohammed A. Mutar, Daniele Ettore Otera and Hasan A. Khawwan
Mathematics 2025, 13(2), 229; https://doi.org/10.3390/math13020229 - 11 Jan 2025
Viewed by 550
Abstract
An edge-coloring σ of a connected graph G is called rainbow if there exists a rainbow path connecting any pair of vertices. In contrast, σ is monochromatic if there is a monochromatic path between any two vertices. Some graphs can admit a coloring [...] Read more.
An edge-coloring σ of a connected graph G is called rainbow if there exists a rainbow path connecting any pair of vertices. In contrast, σ is monochromatic if there is a monochromatic path between any two vertices. Some graphs can admit a coloring which is simultaneously rainbow and monochromatic; for instance, any coloring of Kn is rainbow and monochromatic. This paper refers to such a coloring as dual coloring. We investigate dual coloring on various graphs and raise some questions about the sufficient conditions for connected graphs to be dual connected. Full article
(This article belongs to the Special Issue Geometric Topology and Differential Geometry with Applications)
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