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Axioms
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Advances in Geometry and Its Applications
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Advances in Geometry and Its Applications

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

Deadline for manuscript submissions: closed (31 December 2025) | Viewed by 8294

Special Issue Editors

Dr. Nenad O. Vesić

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Guest Editor
Mathematical Institute, Serbian Academy of Sciences and Arts, 11000 Belgrade, Serbia
Interests: differential geometry of mappings; generalized Riemannian spaces and generalized affine connection spaces applied in physics; applications of differential geometry in multi-criteria decision making
Special Issues, Collections and Topics in MDPI journals
Dr. Rutwig Campoamor-Stursberg

E-Mail Website
Guest Editor
Instituto de Matemática Interdisciplinar, Departamento de Geometría y Topología, Universidad Complutense de Madrid, Plaza de Ciencias 3, E-28040 Madrid, Spain
Interests: real and complex lie algebras and groups; differential forms and distribution theory; contractions and deformations; Casimir invariants; symmetries in physics; representation theory; lie group analysis of differential equations; Lagrangian and Hamiltonian formalism in classical mechanics; integrable and superintegrable systems; symmetry-conditioned perturbation theory; inverse problems in dynamics; supersymmetry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Differential geometry has been applied in mechanics and cosmology. Last year, a paper that generalized the MCDM method in the sense of differential geometry was published (Simjanović, et al., A Novel Fuzzy Analytic Hierarchy Process, Filomat, Vol. 37, No. 11 (2023)). Therefore, this Special Issue addresses the following topics: 1. theoretical differential geometry; 2. Riemannian spaces and cosmology; 3. graphical data analysis in quantum mechanics; and 4. differential geometry based MCDM methods and applications. In part 1, authors should present novel results that have been obtained with respect to tensor calculus. In part 2, novel computations in cosmology should be offered. In part 3, the application of quantum mechanics in chemistry covered by different graphical analyses should be presented. In part 4, I encourage authors to submit papers that address multi-criteria decision-making with respect to the methodology presented in the above-mentioned paper by Simjanović. The aim of this Special Issue is to provide an overview of recent results that provide scope for further novel investigations into various areas of applied mathematics.

Dr. Nenad O. Vesić
Dr. Rutwig Campoamor-Stursberg
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. 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

  • tensor calculus
  • affine connection
  • metric tensor
  • inner product
  • curvature tensor
  • invariants
  • cosmological models
  • energy levels
  • fuzzy numbers and MCDM methods
  • graphical analysis

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Published Papers (6 papers)

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Research

25 pages, 372 KB  
Article
Recognition Geometry
by Jonathan Washburn, Milan Zlatanović and Elshad Allahyarov
Axioms 2026, 15(2), 90; https://doi.org/10.3390/axioms15020090 - 26 Jan 2026
Cited by 2 | Viewed by 1392
Abstract
We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence [...] Read more.
We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map R¯:CR→E is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on Rn, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
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20 pages, 291 KB  
Article
Half-Symmetric Connections of Generalized Riemannian Spaces
by Marko Stefanović, Mića S. Stanković, Ivana Djurišić and Nenad Vesić
Axioms 2025, 14(12), 923; https://doi.org/10.3390/axioms14120923 - 16 Dec 2025
Viewed by 465
Abstract
In this article, we generalize Yano’s concept of a half-symmetric affine connection. With respect to this generalization, we obtain five linearly independent curvature tensors. In the following, we examine which special kinds of affine connections may be the generalized half-symmetric affine connection. At [...] Read more.
In this article, we generalize Yano’s concept of a half-symmetric affine connection. With respect to this generalization, we obtain five linearly independent curvature tensors. In the following, we examine which special kinds of affine connections may be the generalized half-symmetric affine connection. At the end of this work, we generalize the term of Killing’s vector given by Yano to affine Killing, conformal Killing, projective Killing, harmonic, and covariant and contravariant analytic vectors. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
15 pages, 264 KB  
Article
Geometry of Kenmotsu Manifolds via Q-Curvature Tensor and Schouten–Van Kampen Connection
by Mustafa Yıldırım, Selahattin Beyendi, Gülhan Ayar and Nesip Aktan
Axioms 2025, 14(7), 498; https://doi.org/10.3390/axioms14070498 - 26 Jun 2025
Cited by 2 | Viewed by 844
Abstract
This research paper aims to study the Q-curvature tensor on Kenmotsu manifolds endowed with the Schouten–van Kampen connection. Using the Q-curvature tensor, whose trace is the well-known Z-tensor, we characterized Kenmotsu manifolds by introducing the notion of ζ-Q˜ [...] Read more.
This research paper aims to study the Q-curvature tensor on Kenmotsu manifolds endowed with the Schouten–van Kampen connection. Using the Q-curvature tensor, whose trace is the well-known Z-tensor, we characterized Kenmotsu manifolds by introducing the notion of ζ-Q˜ flat and ϕ-Q˜ flat manifolds and novel tensor conditions, such as Q˜(ξ,X)Q˜=0, Q˜(ξ,X)R˜=0, Q˜(ξ,X)C˜=0, Q˜(ξ,X)S˜=0, Q˜(ξ,X)H˜=0, and Q˜(ξ,X)P˜=0, with the Schouten–van Kampen connection. To validate some of our results, we constructed a non-trivial example of Kenmotsu manifolds endowed with the Schouten–van Kampen connection. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
15 pages, 268 KB  
Article
An Optimal Inequality for Warped Product Pointwise Semi-Slant Submanifolds in Complex Space Forms
by Md Aquib
Axioms 2025, 14(3), 213; https://doi.org/10.3390/axioms14030213 - 14 Mar 2025
Cited by 1 | Viewed by 848
Abstract
In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized δ-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these [...] Read more.
In this paper, we utilize advanced optimization techniques on Riemannian submanifolds to establish two distinct inequalities concerning the generalized normalized δ-Casorati curvatures of warped product pointwise semi-slant (WPPSS) submanifolds within complex space forms. We further identify the precise conditions under which these inequalities attain equality, providing valuable insights into their geometric and structural significance. Additionally, we also present results involving harmonic and Hessian functions, revealing a broader connection between curvature properties and analytic functions. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
34 pages, 610 KB  
Article
A Unified Approach to Aitchison’s, Dually Affine, and Transport Geometries of the Probability Simplex
by Giovanni Pistone and Muhammad Shoaib
Axioms 2024, 13(12), 823; https://doi.org/10.3390/axioms13120823 - 25 Nov 2024
Cited by 1 | Viewed by 2088
Abstract
A critical processing step for AI algorithms is mapping the raw data to a landscape where the similarity of two data points is conveniently defined. Frequently, when the data points are compositions of probability functions, the similarity is reduced to affine geometric concepts; [...] Read more.
A critical processing step for AI algorithms is mapping the raw data to a landscape where the similarity of two data points is conveniently defined. Frequently, when the data points are compositions of probability functions, the similarity is reduced to affine geometric concepts; the basic notion is that of the straight line connecting two data points, defined as a zero-acceleration line segment. This paper provides an axiomatic presentation of the probability simplex’s most commonly used affine geometries. One result is a coherent presentation of gradient flow in Aichinson’s compositional data, Amari’s information geometry, the Kantorivich distance, and the Lagrangian optimization of the probability simplex. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
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17 pages, 377 KB  
Article
Isoptic Point of the Complete Quadrangle
by Ema Jurkin, Marija Šimić Horvath and Vladimir Volenec
Axioms 2024, 13(6), 349; https://doi.org/10.3390/axioms13060349 - 24 May 2024
Cited by 1 | Viewed by 1479
Abstract
In this paper, we study the complete quadrangle. We started this investigation in a few of our previous papers. In those papers and here, the rectangular coordinates are used to enable us to prove the properties of the rich geometry of a quadrangle [...] Read more.
In this paper, we study the complete quadrangle. We started this investigation in a few of our previous papers. In those papers and here, the rectangular coordinates are used to enable us to prove the properties of the rich geometry of a quadrangle using the same method. Now, we are focused on the isoptic point of the complete quadrangle ABCD, which is the inverse point to A′,B′,C′, and D′ with respect to circumscribed circles of the triangles BCD, ACD, ABD, and ABC, respectively, where A′,B′,C′, and D′ are isogonal points to A,B,C, and D with respect to these triangles. In studying the properties of the quadrangle regarding its isoptic point, some new results are obtained as well. Full article
(This article belongs to the Special Issue Advances in Geometry and Its Applications)
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