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11 pages, 2345 KiB

Open AccessArticle

(ζ^−m, ζ^m)-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 - 09 Jan 2022

Cited by 1 | Viewed by 1461

Abstract

We introduce the real minimal surfaces family by using the Weierstrass data

(ζ^{- m}, ζ^{m})

for

ζ \in C

,

m \in Z_{\geq 2},

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

(ζ^{- m}, ζ^{m})

for

ζ \in C

,

m \in Z_{\geq 2},

then compute the irreducible algebraic surfaces of the surfaces family in three-dimensional Euclidean space

E^{3}

. In addition, we propose that family has a degree number (resp., class number)

2 m (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

Open AccessArticle

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 2474

Abstract

We consider the Enneper family of real maximal surfaces via Weierstrass data

(1, ζ^{m})

for

ζ \in C

,

m \in Z_{\geq 1}

. 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, ζ^{m})

for

ζ \in C

,

m \in Z_{\geq 1}

. We obtain the irreducible surfaces of the family in the three dimensional Minkowski space

E^{2, 1}

. Moreover, we propose that the family has degree

{(2 m + 1)}^{2}

(resp., class

2 m (2 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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11 pages, 1025 KiB

Open AccessArticle

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 1728

Abstract

We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space

E^{4}

. We find the Gauss map of helicoidal hypersurface in

E^{4}

. 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

E^{4}

. We find the Gauss map of helicoidal hypersurface in

E^{4}

. 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

Open AccessFeature PaperArticle

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 1574

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

Open AccessFeature PaperArticle

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 1892 | 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

Open AccessArticle

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 1422

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

Open AccessArticle

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 - 01 Apr 2021

Viewed by 1577

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

Open AccessArticle

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 - 09 Jan 2021

Cited by 2 | Viewed by 1454

Abstract

We give a simple proof of the Chen inequality for the Chen invariant

\underset{k terms}{\underset{︸}{δ (2, \dots, 2)}}

of submanifolds in Riemannian space forms. Full article

(This article belongs to the Special Issue Applications of Differential Geometry II)

33 pages, 1668 KiB

Open AccessArticle

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 1854

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

Jump to: Research, Other

35 pages, 443 KiB

Open AccessReview

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 1375

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

Jump to: Research, Review

2 pages, 191 KiB

Open AccessCorrection

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 1182

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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