Special Issue "Pseudo-Riemannian Metrics and Applications"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 April 2020).

Special Issue Editor

Prof. Dr. Giovanni Calvaruso
Website
Guest Editor
Dipartimento di Matematica e Fisica 'E. De Giorgi', Università del Salento, Prov. Lecce-Arnesano, 73100 Lecce, Italy
Interests: differential geometry; Lie groups; global analysis; mathematical physics

Special Issue Information

Dear Colleagues,

Pseudo-Riemannian metrics are ubiquitous in differential geometry and its applications to theoretical physics. The study of the geometry of an n-dimensional manifold, once an inner product is assigned to the tangent space of each point, started with the revolutionary work of Riemann in the middle of the 19th century. In the first decades of the 20th century, the mathematical formulation of Einstein’s theory of relativity gave an exceptional impulse to the study of nondegenerate metrics. Since then, the relevance and applications of pseudo-Riemannian metrics has grown steadily. Among the recent achievements in this area, we can mention Perelman’s proof of the Poincaré Conjecture as an example.

The purpose of this Special Issue is to collect original and survey papers concerning relevant state-of-the-art results on pseudo-Riemannian metrics and their applications to Physics. A non-exhaustive list of topics includes homogeneous pseudo-Riemannian manifolds, Lorentzian manifolds, special curvature properties, tangent and unit tangent sphere bundles, Einstein manifolds and Ricci solitons, geodesic and magnetic curves, geometry of submanifolds, pseudo-Riemannian Lie groups, symmetries, special connections and metrics, methods of (pseudo-)Riemannian geometry, and harmonic maps.

Prof. Dr. Giovanni Calvaruso
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 papers will be 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 quarterly 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 1000 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
  • Pseudo-Riemannian metrics
  • Lorentzian metrics
  • Homogeneous manifolds
  • Lie groups
  • Symmetries
  • Geodesics
  • Magnetic curves
  • Curvature
  • Geometry of submanifolds
  • Special connections and metrics
  • Einstein manifolds
  • Ricci solitons
  • Harmonic maps
  • Applications to physics

Published Papers (4 papers)

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Research

Open AccessArticle
Reduction of Homogeneous Pseudo-Kähler Structures by One-Dimensional Fibers
Axioms 2020, 9(3), 94; https://doi.org/10.3390/axioms9030094 - 01 Aug 2020
Abstract
We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures [...] Read more.
We study the reduction procedure applied to pseudo-Kähler manifolds by a one dimensional Lie group acting by isometries and preserving the complex tensor. We endow the quotient manifold with an almost contact metric structure. We use this fact to connect pseudo-Kähler homogeneous structures with almost contact metric homogeneous structures. This relation will have consequences in the class of the almost contact manifold. Indeed, if we choose a pseudo-Kähler homogeneous structure of linear type, then the reduced, almost contact homogeneous structure is of linear type and the reduced manifold is of type C5C6C12 of Chinea-González classification. Full article
(This article belongs to the Special Issue Pseudo-Riemannian Metrics and Applications)
Open AccessArticle
Natural Paracontact Magnetic Trajectories on Unit Tangent Bundles
Axioms 2020, 9(3), 72; https://doi.org/10.3390/axioms9030072 - 30 Jun 2020
Abstract
In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base [...] Read more.
In this paper, we study natural paracontact magnetic trajectories in the unit tangent bundle, i.e., those that are associated to g-natural paracontact metric structures. We characterize slant natural paracontact magnetic trajectories as those satisfying a certain conservation law. Restricting to two-dimensional base manifolds of constant Gaussian curvature and to Kaluza–Klein type metrics on their unit tangent bundles, we give a full classification of natural paracontact slant magnetic trajectories (and geodesics). Full article
(This article belongs to the Special Issue Pseudo-Riemannian Metrics and Applications)
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Open AccessArticle
Conformally Flat Siklos Metrics Are Ricci Solitons
Axioms 2020, 9(2), 64; https://doi.org/10.3390/axioms9020064 - 08 Jun 2020
Abstract
We study and solve the Ricci soliton equation for an arbitrary locally conformally flat Siklos metric, proving that such spacetimes are always Ricci solitons. Full article
(This article belongs to the Special Issue Pseudo-Riemannian Metrics and Applications)
Open AccessArticle
Geometric Study of Marginally Trapped Surfaces in Space Forms and Robertson-Walker Spacetimes—An Overview
Axioms 2020, 9(2), 60; https://doi.org/10.3390/axioms9020060 - 24 May 2020
Abstract
A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and [...] Read more.
A marginally trapped surface in a spacetime is a Riemannian surface whose mean curvature vector is lightlike at every point. In this paper we give an up-to-date overview of the differential geometric study of these surfaces in Minkowski, de Sitter, anti-de Sitter and Robertson-Walker spacetimes. We give the general local descriptions proven by Anciaux and his coworkers as well as the known classifications of marginally trapped surfaces satisfying one of the following additional geometric conditions: having positive relative nullity, having parallel mean curvature vector field, having finite type Gauss map, being invariant under a one-parameter group of ambient isometries, being isotropic, being pseudo-umbilical. Finally, we provide examples of constant Gaussian curvature marginally trapped surfaces and state some open questions. Full article
(This article belongs to the Special Issue Pseudo-Riemannian Metrics and Applications)
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