Applications of Differential Geometry II

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

Deadline for manuscript submissions: closed (30 September 2021) | Viewed by 24712

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Guest Editor
Dipartimento di Matematica“G. Peano”, Università di Torino, 8-10124 Torino, Italy
Interests: differential geometry; complex geometry; Lie groups; geometric flows
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Special Issue Information

Dear Colleagues,

This issue is a continuation of our previous Special Issue on “Applications of Differential Geometry”.

The goal of this Special Issue is to explore the multifaceted nature of applications of differential geometry, in relation also to special types of metrics and geometric structures.

Indeed, differential geometry is not only the standard language used to formulate general relativity, but it has also found applications in medical imaging, computer vision and so on.

An article processing charge (APC) applies to papers accepted after peer review. Please contact us if you would like to publish a paper but are experiencing financial constraints. We look forward to your contribution to this Special Issue.

Prof. Dr. Anna Maria Fino
Guest Editor

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Keywords

  • differential geometry
  • general relativity
  • medical imagining
  • computer vision

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Published Papers (11 papers)

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Research

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11 pages, 2345 KiB  
Article
(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space
by Erhan Güler and Ömer Kişi
Axioms 2022, 11(1), 26; https://doi.org/10.3390/axioms11010026 - 9 Jan 2022
Cited by 1 | Viewed by 1962
Abstract
We introduce the real minimal surfaces family by using the Weierstrass data (ζm,ζm) for ζC, mZ2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional [...] Read more.
We introduce the real minimal surfaces family by using the Weierstrass data (ζm,ζm) for ζC, mZ2, then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space E3. In addition, we propose that family has a degree number (resp., class number) 2m(m+1) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c). Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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12 pages, 1517 KiB  
Article
The Algebraic Surfaces of the Enneper Family of Maximal Surfaces in Three Dimensional Minkowski Space
by Erhan Güler
Axioms 2022, 11(1), 4; https://doi.org/10.3390/axioms11010004 - 22 Dec 2021
Cited by 3 | Viewed by 3032
Abstract
We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζC, mZ1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space [...] Read more.
We consider the Enneper family of real maximal surfaces via Weierstrass data (1,ζm) for ζC, mZ1. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space E2,1. Moreover, we propose that the family has degree (2m+1)2 (resp., class 2m(2m+1)) in the cartesian coordinates x,y,z (resp., in the inhomogeneous tangential coordinates a,b,c). Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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11 pages, 1025 KiB  
Article
The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space
by Erhan Güler
Axioms 2021, 10(3), 186; https://doi.org/10.3390/axioms10030186 - 16 Aug 2021
Viewed by 2228
Abstract
We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we [...] Read more.
We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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12 pages, 288 KiB  
Article
Null Darboux Curve Pairs in Minkowski 3-Space
by Jinhua Qian, Mingyu Sun, Pei Yin and Young-Ho Kim
Axioms 2021, 10(3), 142; https://doi.org/10.3390/axioms10030142 - 30 Jun 2021
Cited by 2 | Viewed by 1942
Abstract
Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of [...] Read more.
Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are defined which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented explicitly. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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27 pages, 396 KiB  
Article
Vector Fields and Differential Forms on the Orbit Space of a Proper Action
by Larry Bates, Richard Cushman and Jędrzej Śniatycki
Axioms 2021, 10(2), 118; https://doi.org/10.3390/axioms10020118 - 10 Jun 2021
Cited by 1 | Viewed by 2459 | Correction
Abstract
In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view [...] Read more.
In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
7 pages, 223 KiB  
Article
The χ-Hessian Quotient for Riemannian Metrics
by Pişcoran Laurian-Ioan, Akram Ali, Barbu Cătălin and Ali H. Alkhaldi
Axioms 2021, 10(2), 69; https://doi.org/10.3390/axioms10020069 - 19 Apr 2021
Cited by 1 | Viewed by 1845
Abstract
Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce [...] Read more.
Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One of the main goals of this paper will be to find a link between these two research fields. In this respect, in the present paper, we will introduce and analyze two important quantities in pseudo-Riemannian geometry, namely the H-distorsion and, respectively, the Hessian χ-quotient. This second quantity will be investigated using the Frobenius (Hilbert–Schmidt) norm. Some important examples will be also given, which will prove the validity of the developed theory along the paper. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
30 pages, 2236 KiB  
Article
An Affine Model of a Riemann Surface Associated to a Schwarz–Christoffel Mapping
by Richard Cushman
Axioms 2021, 10(2), 49; https://doi.org/10.3390/axioms10020049 - 1 Apr 2021
Viewed by 1915
Abstract
In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and [...] Read more.
In this paper, we construct an affine model of a Riemann surface with a flat Riemannian metric associated to a Schwarz–Christoffel mapping of the upper half plane onto a rational triangle. We explain the relation between the geodesics on this Riemann surface and billiard motions in a regular stellated n-gon in the complex plane. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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6 pages, 222 KiB  
Article
An Algebraic Inequality with Applications to Certain Chen Inequalities
by Ion Mihai and Radu-Ioan Mihai
Axioms 2021, 10(1), 7; https://doi.org/10.3390/axioms10010007 - 9 Jan 2021
Cited by 2 | Viewed by 1818
Abstract
We give a simple proof of the Chen inequality for the Chen invariant δ(2,,2)k terms of submanifolds in Riemannian space forms. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
33 pages, 1668 KiB  
Article
Shifting Operators in Geometric Quantization
by Richard Cushman and Jędrzej Śniatycki
Axioms 2020, 9(4), 125; https://doi.org/10.3390/axioms9040125 - 28 Oct 2020
Cited by 1 | Viewed by 2318
Abstract
The original Bohr-Sommerfeld theory of quantization did not give operators of transitions between quantum quantum states. This paper derives these operators, using the first principles of geometric quantization. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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Review

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35 pages, 443 KiB  
Review
The Rigging Technique for Null Hypersurfaces
by Manuel Gutiérrez and Benjamín Olea
Axioms 2021, 10(4), 284; https://doi.org/10.3390/axioms10040284 - 29 Oct 2021
Cited by 4 | Viewed by 1806
Abstract
Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null [...] Read more.
Starting from the main definitions, we review the rigging technique for null hypersurfaces theory and most of its main properties. We make some applications to illustrate it. On the one hand, we show how we can use it to show properties of null hypersurfaces, with emphasis in null cones, totally geodesic, totally umbilic, and compact null hypersurfaces. On the other hand, we show the interplay with the ambient space, including its influence in causality theory. Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)

Other

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2 pages, 191 KiB  
Correction
Correction: Bates et al. Vector Fields and Differential Forms on the Orbit Space of a Proper Action. Axioms 2021, 10, 118
by Larry Bates, Richard Cushman and Jędrzej Śniatycki
Axioms 2021, 10(3), 173; https://doi.org/10.3390/axioms10030173 - 30 Jul 2021
Viewed by 1419
Abstract
There was an error in the original article [...] Full article
(This article belongs to the Special Issue Applications of Differential Geometry II)
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