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Axioms
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Theory and Applications: Differential Geometry
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Theory and Applications: Differential Geometry

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

Deadline for manuscript submissions: 31 December 2026 | Viewed by 1332

Special Issue Editor

Prof. Dr. Xiaomin Chen

E-Mail Website
Guest Editor
College of Science, China University of Petroleum (Beijing), Beijing, China
Interests: geometry

Special Issue Information

Dear Colleagues,

Differential geometry, as a core branch of mathematics that explores the properties of curves, surfaces, and higher-dimensional manifolds using calculus and linear algebra, has witnessed remarkable breakthroughs in recent decades. Driven by the rapid development of interdisciplinary fields such as theoretical physics (e.g., general relativity, string theory), computer science (e.g., computer graphics, 3D modeling), and engineering (e.g., robotics, geometric design), differential geometry has transcended traditional theoretical boundaries and become an indispensable tool for solving complex practical problems. Its ability to describe and analyze geometric structures in a quantitative and local-to-global manner has attracted a growing number of researchers from both pure and applied mathematics communities to engage in this dynamic field.

It is my great pleasure to invite you to submit your research papers to this Special Issue of Axioms, entitled “Theory and Applications: Differential Geometry”. This Special Issue is dedicated to creating a high-level academic platform for the dissemination of cutting-edge research findings in differential geometry, with a particular focus on bridging the gap between theoretical advancements and real-world applications. We welcome submissions that not only push the frontiers of fundamental theory in differential geometry but also demonstrate innovative applications in various interdisciplinary domains—provided that the work contributes to the advancement of either theoretical understanding or practical utility in the field of differential geometry.

In this Special Issue, original research articles, comprehensive review papers, and short communications (on emerging research directions) are all welcome. Research areas may include (but are not limited to) the following:

  • Riemannian and pseudo-Riemannian geometry (curvature analysis, geodesic flows, isometric embeddings, and their connections to general relativity);
  • Symplectic and contact geometry (symplectic invariants, Hamiltonian systems, and applications in classical mechanics and control theory);
  • Differential geometry of submanifolds (minimal surfaces, constant mean curvature surfaces, and applications in materials science and surface design);
  • Discrete differential geometry (discrete manifolds, discrete curvatures, and applications in computer graphics, mesh processing, and digital geometry processing);
  • Geometric analysis (elliptic and parabolic PDEs on manifolds, harmonic maps, and applications in image processing and machine learning);
  • Differential geometry in robotics (kinematic modeling of robotic arms, path planning based on Riemannian metrics, and geometric control of mechanical systems).

Prof. Dr. Xiaomin Chen
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 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. 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 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

  • differential geometry
  • riemannian geometry
  • symplectic geometry
  • discrete differential geometry
  • geometric analysis
  • differential geometry in robotics

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (4 papers)

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Research

34 pages, 2258 KB  
Article
Spline-Based Smoothing of Noisy Discrete Curves in the Frenet–Serret Framework: Sensitivity Analysis of Curvature and Torsion Estimation via CSI and TSI Indices for Analytically Defined Space Curves
by Gülden Altay Suroğlu, Şeyma Firdevs Hızal and Hasan Bulut
Axioms 2026, 15(5), 365; https://doi.org/10.3390/axioms15050365 - 14 May 2026
Abstract
This study investigates the robustness of Frenet–Serret curvature (κ) and torsion (τ) estimates derived from noisy discretely-sampled three-dimensional space curves, with emphasis on the comparative performance of cubic spline and cubic Hermite interpolation methods. Accurate estimation of these geometric [...] Read more.
This study investigates the robustness of Frenet–Serret curvature (κ) and torsion (τ) estimates derived from noisy discretely-sampled three-dimensional space curves, with emphasis on the comparative performance of cubic spline and cubic Hermite interpolation methods. Accurate estimation of these geometric invariants is essential for reliable analysis of curves arising in signal processing and shape reconstruction; yet, the higher-order derivatives required for their computation exhibit pronounced sensitivity to measurement noise. We examine curves constructed through a Hilbert transform-based parameterization of the form r(t)=X(t),A(t)sinϕ(t),g(t), where discrete samples are contaminated with additive white Gaussian noise at varying signal-to-noise ratios. Reconstruction is performed using cubic spline interpolation, which ensures global C2 continuity, as well as cubic Hermite spline interpolation, which provides C1 continuity with local tangent control. Frenet frame computations are then applied via regularized finite difference schemes. To characterize noise amplification theoretically, we derive the Curvature Stability Index (CSI) and Torsion Stability Index (TSI) as first-order variance bounds under the delta method. While these indices formalize the derivative-order dependence of noise sensitivity, Monte Carlo simulations reveal that empirical variance exceeds theoretical predictions by factors of 104 to 106, indicating dominance of nonlinear error propagation. Nevertheless, the indices establish that torsion instability arises fundamentally from third-order derivative structure rather than ground-truth magnitude. Numerical experiments across three geometric regimes constant-invariant helices, variable-curvature helices, and planar curves with identically zero torsion demonstrate that the ratio of the torsion root mean square error to curvature root mean square error consistently ranges from 6.5 to 9.8. This disparity persists even in the degenerate planar case, where τ≡0 analytically, confirming that torsion sensitivity is an intrinsic property of the Frenet–Serret formulation. Across all configurations, cubic spline reconstruction yields lower Monte Carlo mean RMSE and reduced empirical variance compared to Hermite spline, providing superior stability for derivative-based invariant estimation. Full article
(This article belongs to the Special Issue Theory and Applications: Differential Geometry)
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14 pages, 266 KB  
Article
An Algebraic Approach to Geodesics via Mobi Spaces
by Jorge Pereira Fatelo and Nelson Martins-Ferreira
Axioms 2026, 15(5), 352; https://doi.org/10.3390/axioms15050352 - 9 May 2026
Viewed by 165
Abstract
Mobi spaces were introduced by the authors as a possible algebraic axiomatization of spaces in which any two points are connected by a geodesic path. Previous work has focused primarily on the algebraic properties of these structures; here, we return to the original [...] Read more.
Mobi spaces were introduced by the authors as a possible algebraic axiomatization of spaces in which any two points are connected by a geodesic path. Previous work has focused primarily on the algebraic properties of these structures; here, we return to the original geometric motivation. We present new characterizations of mobi spaces inspired by the important class of smooth manifolds that arise as open subsets of Euclidean (n)-space endowed with a Riemannian metric. We show that such manifolds satisfy the axioms of a mobi space, thereby providing a broad family of natural geometric examples. Full article
(This article belongs to the Special Issue Theory and Applications: Differential Geometry)
21 pages, 1506 KB  
Article
A Unified Rotation-Minimizing Darboux Framework for Curves and Relativistic Ruled Surfaces in Minkowski Three-Space
by Mona Bin-Asfour, Ghaliah Alhamzi, Emad Solouma and Sayed Saber
Axioms 2026, 15(3), 207; https://doi.org/10.3390/axioms15030207 - 11 Mar 2026
Viewed by 339
Abstract
We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space E13. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior [...] Read more.
We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space E13. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior characteristic of RM frames, yielding a natural geometric description of curves in a Lorentzian environment. For unit speed non-null curves, the governing equations of the RM Darboux frame are derived, and precise connections between the RM curvature functions and the classical Frenet and Darboux invariants are obtained, thereby elucidating the geometric significance of RM curvatures in Lorentzian geometry. Within this setting, multiple classes of ruled surfaces are generated using RM Darboux frame vector fields. Necessary and sufficient conditions for developability, minimality, and flatness are formulated exclusively in terms of RM curvature quantities. The role of the causal character of the generating curve is analyzed in detail, revealing distinct geometric behaviors for space-like and time-like cases. These findings indicate that the RM Darboux framework constitutes a flexible and effective approach for modeling curve-induced surface geometries in Minkowski space, with potential relevance to relativistic kinematics, world sheet constructions, and geometric problems arising in mathematical physics. Full article
(This article belongs to the Special Issue Theory and Applications: Differential Geometry)
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18 pages, 324 KB  
Article
Curvature Bounds and Casorati Pinching for Submanifolds in Kähler Product Manifolds
by Md Aquib, Ibrahim Al-Dayel, Mohd Aslam and Oğuzhan Bahadır
Axioms 2026, 15(3), 168; https://doi.org/10.3390/axioms15030168 - 27 Feb 2026
Viewed by 319
Abstract
In this paper, we establish sharp pinching inequalities that relate the generalized δ-Casorati curvatures to the normalized scalar curvature of submanifolds immersed in Kähler product manifolds endowed with a quarter-symmetric metric connection. The results are obtained for a broad range of geometric [...] Read more.
In this paper, we establish sharp pinching inequalities that relate the generalized δ-Casorati curvatures to the normalized scalar curvature of submanifolds immersed in Kähler product manifolds endowed with a quarter-symmetric metric connection. The results are obtained for a broad range of geometric configurations, encompassing several important classes of submanifolds. Moreover, we prove that the derived inequalities are optimal by completely characterizing the submanifolds for which equality holds, showing that these cases correspond precisely to invariantly quasi-umbilical submanifolds with trivial normal connection. Full article
(This article belongs to the Special Issue Theory and Applications: Differential Geometry)
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