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Axioms
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Advances in Differential Geometry and Mathematical Physics
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Advances in Differential Geometry and Mathematical Physics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 30 December 2024 | Viewed by 2696

Special Issue Editor

Dr. David D. McNutt

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, UiT The Arctic University of Norway, 9019 Tromsø, Norway
Interests: alternative gravity theories; Cartan invariants; Pseudo-Riemannian geometry; teleparallel geometry; classification of manifolds

Special Issue Information

Dear Colleagues,

I am acting as a guest editor for a Special Issue on advances in differential geometry and mathematical physics in MDPI’s journal Axioms. Our intention with this Special Issue is to focus on new and interesting applications of differential geometry inspired by general relativity, its modifications and alternative gravity theories.

In particular, we would like to provide an opportunity to present recent developments in mathematical physics that incorporate geometries beyond curvature based Lorentzian geometries. This Special Issue will address the following non-exhaustive list of topics:

  • Symmetry methods.
  • Conformal symmetries.
  • Invariants associated with geometries.
  • Mathematical aspects of solutions to particular gravity theories.
  • Applications of pseudo-Riemannian geometries, teleparallel geometries, symmetric teleparallel geometries, Einstein–Cartan geometries or Finsler geometries to mathematical physics.

In addition to the above, any topic that relates to the application of differential geometry in mathematical physics is welcome.

We hope that this initiative will be attractive to experts in the field of mathematical physics who are exploring new ways to apply differential geometry to problems in mathematical physics. We encourage you to submit your current research or reviews to be included in the Special Issue.

Dr. David D. McNutt
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. 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

  • Pseudo-Riemannian geometry
  • teleparallel geometry
  • Riemann–Cartan geometry
  • symmetries
  • invariants
  • black holes
  • conformal symmetries
  • alternative theories of gravity

Published Papers (3 papers)

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Research

26 pages, 5396 KiB  
Article
Double-Step Shape Invariance of Radial Jacobi-Reference Potential and Breakdown of Conventional Rules of Supersymmetric Quantum Mechanics
by Gregory Natanson
Axioms 2024, 13(4), 273; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13040273 - 19 Apr 2024
Viewed by 307
Abstract
The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two [...] Read more.
The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)
10 pages, 250 KiB  
Article
Lax Pairs for the Modified KdV Equation
by Georgy I. Burde
Axioms 2024, 13(2), 121; https://doi.org/10.3390/axioms13020121 - 14 Feb 2024
Viewed by 878
Abstract
Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its [...] Read more.
Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its possible applications to other types of evolution equations are discussed. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)
16 pages, 306 KiB  
Article
Intrinsic Geometric Structure of Subcartesian Spaces
by Richard Cushman and Jędrzej Śniatycki
Axioms 2024, 13(1), 9; https://0-doi-org.brum.beds.ac.uk/10.3390/axioms13010009 - 22 Dec 2023
Viewed by 767
Abstract
Every subset S of a Cartesian space Rd, endowed with differential structure C∞(S) generated by restrictions to S of functions in C∞(Rd), has a canonical partition M(S) by [...] Read more.
Every subset S of a Cartesian space Rd, endowed with differential structure C∞(S) generated by restrictions to S of functions in C∞(Rd), has a canonical partition M(S) by manifolds, which are orbits of the family X(S) of all derivations of C∞(S) that generate local one-parameter groups of local diffeomorphisms of S. This partition satisfies the frontier condition, Whitney’s conditions A and B. If M(S) is locally finite, then it satisfies all definitions of stratification of S. This result extends to Hausdorff locally Euclidean differential spaces. The partition M(S) of a subcartesian space S by smooth manifolds provides a measure for the applicability of differential geometric methods to the study of the geometry of S. If all manifolds in M(S) are single points, we cannot expect differential geometry to be an effective tool in the study of S. On the other extreme, if M(S) contains only one manifold M, then the subcartesian space S is a manifold, S=M, and it is a natural domain for differential geometric techniques. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)
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