10th Anniversary of Axioms: Geometry and Topology

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

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 7282

Special Issue Editor


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Guest Editor
1. School of Engineering, IT and Physical Sciences, Federation University Australia, Ballarat, VIC 3353, Australia
2. Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, VIC 3086, Australia
Interests: topological groups; especially locally compact groups; pro-Lie groups; topological algebra; topological vector spaces; Banach spaces; topology; group theory; functional analysis; universal algebra; transcendental number theory; numerical geometry; history of mathematics; information technology security; health informatics; international education; university education; online education; social media in the teaching of mathematics; stock market prediction; managing scholarly journals
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Special Issue Information

Dear Colleagues,

This year, Axioms is celebrating its 10th anniversary from the launch of its inaugural issue in 2012. We greatly appreciate the efforts and support of our authors, reviewers, readers and editors over the last ten years. To mark this significant milestone and commemorate the journal’s achievements throughout the years, we are organizing a Special Issue entitled "10th Anniversary of Axioms: Geometry and Topology”.

This Special Issue will present a collection of high-quality original research articles and surveys in the fields of geometry and topology. Articles considering significant open questions are encouraged. Of particular interest are contributions addressing topics including, but not limited to: algebraic geometry; analytic geometry; computational geometry; differential geometry; discrete geometry; symplectic geometry; hyperbolic geometry; finite geometry; geometric group theory;  affine geometry; general topology; low-dimensional topology; algebraic topology; differential topology; geometric topology; topological groups; Lie groups; topological algebra; geometry of Banach spaces;  and topological vector spaces. Papers detailing applications of topology and geometry to art, architecture, computer science, physics, astronomy, set theory, differential equations, and string theory are especially encouraged.

In this Special Issue on geometry and topology, we seek to address topics which fall into these diverse categories. Original articles reporting recent progress and survey articles are also sought. Authors are encouraged to contemplate interesting open questions.

Prof. Dr. Sidney A. Morris
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

  • topology
  • algebraic topology
  • differential topology
  • topological groups
  • lie group
  • differential geometry
  • algebraic geometry
  • discrete geometry

Published Papers (5 papers)

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Research

11 pages, 302 KiB  
Article
The Stereographic Projection in Topological Modules
by Francisco Javier García-Pacheco
Axioms 2023, 12(2), 225; https://doi.org/10.3390/axioms12020225 - 20 Feb 2023
Cited by 2 | Viewed by 1146
Abstract
The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a topological R-module M such that 0int(A). If there exists a continuous functional m*:MR [...] Read more.
The stereographic projection is constructed in topological modules. Let A be an additively symmetric closed subset of a topological R-module M such that 0int(A). If there exists a continuous functional m*:MR in the dual module M*, an invertible sU(R) and an element a in the topological boundary bd(A) of A in such a way that m*1({s})int(A)=, am*1({s})bd(A), and s+m*bd(A)\{a}U(R), then the following function ba+2s(m*(b)+s)1(b+a), from bd(A)\{a} to (m*)1({s}), is a well-defined stereographic projection (also continuous if multiplicative inversion is continuous on R). Finally, we provide sufficient conditions for the previous stereographic projection to become a homeomorphism. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Geometry and Topology)
10 pages, 282 KiB  
Article
Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field
by Mancho Manev
Axioms 2023, 12(1), 44; https://doi.org/10.3390/axioms12010044 - 1 Jan 2023
Cited by 1 | Viewed by 1281
Abstract
A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case [...] Read more.
A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case in which the potential is a torse-forming vector field of constant length on the vertical distribution determined by the Reeb vector field is studied. In this way, manifolds from one of the main classes of the studied manifolds are obtained. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the given B-metric. An explicit five-dimensional example of a Lie group is given, which is characterized in relation to the obtained results. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Geometry and Topology)
16 pages, 344 KiB  
Article
Schwarzschild Spacetimes: Topology
by Demeter Krupka and Ján Brajerčík
Axioms 2022, 11(12), 693; https://doi.org/10.3390/axioms11120693 - 4 Dec 2022
Cited by 2 | Viewed by 1698
Abstract
This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group [...] Read more.
This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3{(0,0,0)}) and S1×(R3{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S1×(R3{(0,0,0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Geometry and Topology)
8 pages, 281 KiB  
Article
Categorically Closed Unipotent Semigroups
by Taras Banakh and Myroslava Vovk
Axioms 2022, 11(12), 682; https://doi.org/10.3390/axioms11120682 - 29 Nov 2022
Cited by 1 | Viewed by 1048
Abstract
Let C be a class of T1 topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in any topological semigroup YC that contains X as a discrete subsemigroup; X is injectively [...] Read more.
Let C be a class of T1 topological semigroups, containing all Hausdorff zero-dimensional topological semigroups. A semigroup X is C-closed if X is closed in any topological semigroup YC that contains X as a discrete subsemigroup; X is injectively C-closed if for any injective homomorphism h:XY to a topological semigroup YC the image h[X] is closed in Y. A semigroup X is unipotent if it contains a unique idempotent. It is proven that a unipotent commutative semigroup X is (injectively) C-closed if and only if X is bounded and nonsingular (and group-finite). This characterization implies that for every injectively C-closed unipotent semigroup X, the center Z(X) is injectively C-closed. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Geometry and Topology)
15 pages, 332 KiB  
Article
Chaos in Topological Modules
by Francisco Javier García-Pacheco
Axioms 2022, 11(10), 526; https://doi.org/10.3390/axioms11100526 - 2 Oct 2022
Cited by 1 | Viewed by 1309
Abstract
Chaotic and pathological phenomena in topological modules are studied in this manuscript. In particular, constructions of noncontinuous linear functionals are provided for a wide variety of topological modules. In addition, constructions of balanced and absorbing sets which are not neighborhoods of zero are [...] Read more.
Chaotic and pathological phenomena in topological modules are studied in this manuscript. In particular, constructions of noncontinuous linear functionals are provided for a wide variety of topological modules. In addition, constructions of balanced and absorbing sets which are not neighborhoods of zero are also given in an extensive class of topological modules. Finally, we construct a linearly open set with empty interior in a large amount of topological modules. All these constructions are related to each other. Prior to developing all these results, we provide an axiomatization of the topological concept of limit by introducing the limit operators in a similar context as hull operators or closure operators are defined. Full article
(This article belongs to the Special Issue 10th Anniversary of Axioms: Geometry and Topology)
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