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Mathematics
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New Trends in Differential Geometry and Geometric Analysis
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New Trends in Differential Geometry and Geometric Analysis

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

Deadline for manuscript submissions: 31 August 2026 | Viewed by 4111

Special Issue Editors

Dr. Marija S. Najdanović

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Guest Editor
Faculty of Sciences and Mathematics, University of Priština in Kosovska Mitrovica, 38220 Kosovska Mitrovica, Serbia
Interests: differential geometry of curves and surfaces; infinitesimal bending; generalized Riemannian spaces; geometric knot theory; applications of geometry to natural processes
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ljubica Velimirovic

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Guest Editor
Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, 18 000 Niš, Serbia
Interests: differential geometry; analytical geometry; foundations of geometry; descriptive geometry; geometry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue aims to highlight recent developments and directions in differential geometry and geometric analysis, two fields that lie at the heart of many significant advances in mathematics and its applications. These areas have seen remarkable progress through the interplay of geometric intuition and analytical techniques, yielding deep insights into the structure and behavior of spaces and mappings.

We welcome contributions on a wide range of topics, including differential geometry of curves, surfaces and manifolds, Riemannian and complex geometry, spaces with non-symmetric affine connection, Minkowski spaces, dual spaces, shape and energy of geometric objects, geometric knot theory, geometric flows, geometric partial differential equations, variational methods in geometry, and applications to related fields such as topology and mathematical physics.

All related topics within the broader scope of differential geometry and geometric analysis are welcome for this Special Issue.

Dr. Marija S. Najdanović
Prof. Dr. Ljubica Velimirovic
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 250 words) can be sent to the Editorial Office for assessment.

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

  • differential geometry
  • geometric analysis
  • infinitesimal deformations
  • curves and surfaces
  • manifolds
  • knots
  • shape and energy of geometric objects
  • Riemannian geometry and generalizations
  • minkowski spaces
  • dual spaces
  • geometric flows
  • geometric partial differential equations
  • minimal surfaces
  • variational methods
  • global analysis.

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Published Papers (6 papers)

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Research

21 pages, 729 KB  
Article
Geometric Analysis of Soliton Surfaces via Generalized Bishop Frame Field of Type-C Associated with Betchov–Da Rios Equation in E4
by Yanlin Li, Ahmet Kazan and Mustafa Altın
Mathematics 2026, 14(8), 1290; https://doi.org/10.3390/math14081290 - 13 Apr 2026
Viewed by 173
Abstract
In this study, using the generalized Bishop frame field of type-C in four-dimensional Euclidean space, we investigate the geometric properties of a soliton surface M=M(x,y) associated with the Betchov–Da Rios equation. Using a unit speed [...] Read more.
In this study, using the generalized Bishop frame field of type-C in four-dimensional Euclidean space, we investigate the geometric properties of a soliton surface M=M(x,y) associated with the Betchov–Da Rios equation. Using a unit speed x-parameter curve M=M(x,y), for all y, we derive the derivative formulas for the generalized Bishop frame field of type-C. We obtain two fundamental geometric invariants of the soliton surface, k and h, characterizing the points of the surface, as well as a few other significant invariants, including Gaussian curvature, a mean curvature vector and Gaussian torsion. We use these surface invariants to prove a collection of theorems that describe the circumstances in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal). Additionally, we provide a theorem that describes the B-DR soliton surface’s curvature ellipse in relation to the generalized Bishop frame field of type-C in E4. Finally, we construct a foundational example to support our theoretical results and demonstrate the construction of the generalized Bishop frame field of type-C. Full article
(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)
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12 pages, 370 KB  
Article
Surfaces of Revolution with Constant Mean Curvature in Galilean 3-Space
by İsmet Gölgeleyen, Yusuf Yaylı and Elif Yaren Bulgan
Mathematics 2026, 14(6), 1066; https://doi.org/10.3390/math14061066 - 21 Mar 2026
Viewed by 245
Abstract
Revolution surfaces with zero mean curvature in the Galilean 3-space have been extensively studied in the literature. However, revolution surfaces with non-zero constant mean curvature in this geometric setting have not yet been investigated in a systematic way. In this paper, we address [...] Read more.
Revolution surfaces with zero mean curvature in the Galilean 3-space have been extensively studied in the literature. However, revolution surfaces with non-zero constant mean curvature in this geometric setting have not yet been investigated in a systematic way. In this paper, we address this gap by studying surfaces of revolution in the Galilean 3-space with constant mean curvature. We derive the necessary and sufficient differential conditions for such surfaces and obtain explicit parametrizations of the corresponding families. The results extend the theory beyond the minimal case and reveal geometric features that arise from the degenerate nature of the Galilean metric. Several examples are presented to illustrate the obtained surfaces and to emphasize the qualitative differences between minimal and non-minimal constant mean curvature configurations. Full article
(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)
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34 pages, 495 KB  
Article
Rigidity and Toledo Invariant for Spin*(8)-Higgs Bundles
by Álvaro Antón-Sancho
Mathematics 2026, 14(2), 358; https://doi.org/10.3390/math14020358 - 21 Jan 2026
Viewed by 307
Abstract
In this paper, we study Spin*(8)-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on SO*(8). The group Spin*(8) is exceptional among classical real [...] Read more.
In this paper, we study Spin*(8)-Higgs bundles over compact Riemann surfaces, extending the work of Bradlow, García-Prada, and Gothen on SO*(8). The group Spin*(8) is exceptional among classical real forms, as its complexification Spin(8,C) admits triality, an outer automorphism of order 3, but triality does not preserve the real form Spin*(8). We establish the Toledo bound |τ|≤4(g−1) for semistable Spin*(8)-Higgs bundles and characterize maximal bundles through rigidity theorems. We prove that the moduli space of maximal bundles fibers over the SO*(8) moduli space with discrete fibers parametrized by spin structures, and has a dimension of 15(g−1), one less than expected. Using Morse theory, we establish connectedness of moduli spaces for τ=0 and maximal |τ|. Via the non-abelian Hodge correspondence, our results yield connectedness theorems for character varieties of surface group representations into Spin*(8). We analyze how triality determines the decomposition of the isotropy representation despite not acting on the real form. Full article
(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)
15 pages, 337 KB  
Article
Developable Ruled Surfaces with Constant Mean Curvature Along a Curve
by İsmet Gölgeleyen, Yusuf Yaylı and Elif Yaren Bulgan
Mathematics 2026, 14(2), 234; https://doi.org/10.3390/math14020234 - 8 Jan 2026
Cited by 1 | Viewed by 459
Abstract
In this work, we study developable ruled surfaces with constant mean curvature along a curve. The mean curvature of developable ruled surfaces generated by indicatrix curves is calculated. The analysis is first carried out in Euclidean three-space and then extended to Lorentz space. [...] Read more.
In this work, we study developable ruled surfaces with constant mean curvature along a curve. The mean curvature of developable ruled surfaces generated by indicatrix curves is calculated. The analysis is first carried out in Euclidean three-space and then extended to Lorentz space. For both geometries, we derive the necessary and sufficient conditions under which the developable ruled surfaces exhibit constant mean curvature. In addition, we calculate the mean curvature of the surface using time-like and space-like curves. Later, we give a sufficient condition for the mean curvature of a developable surface to be constant along a striction curve. Finally, we give some examples in Euclidean and Lorentz spaces and present computational examples. Full article
(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)
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13 pages, 304 KB  
Article
Weak Nearly S- and Weak Nearly C-Manifolds
by Vladimir Rovenski
Mathematics 2025, 13(19), 3169; https://doi.org/10.3390/math13193169 - 3 Oct 2025
Viewed by 601
Abstract
The recent interest in geometers in the f-structures of K. Yano is motivated by the study of the dynamics of contact foliations, as well as their applications in theoretical physics. Weak metric f-structures on a smooth manifold, recently introduced by the [...] Read more.
The recent interest in geometers in the f-structures of K. Yano is motivated by the study of the dynamics of contact foliations, as well as their applications in theoretical physics. Weak metric f-structures on a smooth manifold, recently introduced by the author and R. Wolak, open a new perspective on the theory of classical structures. In this paper, we define structures of this kind, called weak nearly S- and weak nearly C-structures, study their geometry, e.g., their relations to Killing vector fields, and characterize weak nearly S- and weak nearly C-submanifolds in a weak nearly Kähler manifold. Full article
(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)
19 pages, 301 KB  
Article
Geometric and Structural Properties of Indefinite Kenmotsu Manifolds Admitting Eta-Ricci–Bourguignon Solitons
by Md Aquib, Oğuzhan Bahadır, Laltluangkima Chawngthu and Rajesh Kumar
Mathematics 2025, 13(12), 1965; https://doi.org/10.3390/math13121965 - 14 Jun 2025
Viewed by 697
Abstract
This paper undertakes a detailed study of η-Ricci–Bourguignon solitons on ϵ-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic η-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying [...] Read more.
This paper undertakes a detailed study of η-Ricci–Bourguignon solitons on ϵ-Kenmotsu manifolds, with particular focus on three special types of Ricci tensors: Codazzi-type, cyclic parallel and cyclic η-recurrent tensors that support such solitonic structures. We derive key curvature conditions satisfying Ricci semi-symmetric (R·E=0), conharmonically Ricci semi-symmetric (C(ξ,βX)·E=0), ξ-projectively flat (P(βX,βY)ξ=0), projectively Ricci semi-symmetric (L·P=0) and W5-Ricci semi-symmetric (W(ξ,βY)·E=0), respectively, with the admittance of η-Ricci–Bourguignon solitons. This work further explores the role of torse-forming vector fields and provides a thorough characterization of ϕ-Ricci symmetric indefinite Kenmotsu manifolds admitting η-Ricci–Bourguignon solitons. Through in-depth analysis, we establish significant geometric constraints that govern the behavior of these manifolds. Finally, we construct explicit examples of indefinite Kenmotsu manifolds that satisfy the η-Ricci–Bourguignon solitons equation, thereby confirming their existence and highlighting their unique geometric properties. Moreover, these solitonic structures extend soliton theory to indefinite and physically meaningful settings, enhance the classification and structure of complex geometric manifolds by revealing how contact structures behave under advanced geometric flows and link the pure mathematical geometry to applied fields like general relativity. Furthermore, η-Ricci–Bourguignon solitons provide a unified framework that deepens our understanding of geometric evolution and structure-preserving transformations. Full article
(This article belongs to the Special Issue New Trends in Differential Geometry and Geometric Analysis)
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