Research

13 pages, 326 KiB

Open AccessArticle

Curve-Surface Pairs on Embedded Surfaces and Involute D-Scroll of the Curve-Surface Pair in E³

by Filiz Ertem Kaya and Süleyman Şenyurt

Symmetry 2024, 16(3), 323; https://doi.org/10.3390/sym16030323 - 07 Mar 2024

Viewed by 613

Willmore defined embedded surfaces on

f : S \to E^{3}

, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of

E^{3}

, inducing a Riemannian structure on

f (S)

. The expression analogous [...] Read more.

Willmore defined embedded surfaces on

f : S \to E^{3}

, which is the embedding of S into Euclidean 3-space. He investigated the Euclidean metric of

E^{3}

, inducing a Riemannian structure on

f (S)

. The expression analogous to the left-hand member of the curvature K is replaced by the mean curvature

H^{2}

on

f (S) .

Our aim is to observe the Gaussian and mean curvatures of curve–surface pairs using embedded surfaces in different curve–surface pairs and to define some developable operations on their curve–surface pairs. We also investigate the embedded surfaces using the Willmore method. We first recall the Darboux curve–surface and derive the new characterizations. This curve–surface pair is called the osculating Darboux curve–surface if its position vector always lies in the osculating Darboux plane spanned by a Darboux frame. Thus, we observed an osculating Darboux curve–surface pair. We also obtained the

\tilde{D}

-scroll of the curve–surface pair and involute

\tilde{D}

-scroll of the curve–surface pair with some differential geometric elements and found

{\tilde{D}}_{(α, M)} (s)

and

{\tilde{D^{*}}}_{(α, M)} (s)

-scrolls of the curve–surface pair

(α, M)

. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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17 pages, 1290 KiB

Open AccessArticle

A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

by Gudrun Albrecht, Esmeralda Mainar, Juan Manuel Peña and Beatriz Rubio

Symmetry 2023, 15(11), 2041; https://doi.org/10.3390/sym15112041 - 10 Nov 2023

Viewed by 647

Abstract

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space [...] Read more.

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space

U_{4} (I_{α}) = span {1, cos t, sin t, cos 2 t, sin 2 t}

defined on compact intervals

I_{α} = [0, α]

, where

α

is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space

U_{4} (I_{α})

to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as

α \to 0

. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of

C^{(3}

Hermite interpolation problems. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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16 pages, 295 KiB

Open AccessArticle

Proposed Theorems on the Lifts of Kenmotsu Manifolds Admitting a Non-Symmetric Non-Metric Connection (NSNMC) in the Tangent Bundle

by Rajesh Kumar, Lalnunenga Colney and Mohammad Nazrul Islam Khan

Symmetry 2023, 15(11), 2037; https://doi.org/10.3390/sym15112037 - 09 Nov 2023

Cited by 2 | Viewed by 1003

Abstract

The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of [...] Read more.

The main aim of the proposed paper is to investigate the lifts of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We investigate several properties of the lifts of the curvature tensor, the conformal curvature tensor, and the conharmonic curvature tensor of Kenmotsu manifolds that admit NSNMC in the tangent bundle. We also study and discover that the lift of the Kenmotsu manifold that admit NSNMC is regular in the tangent bundle. Additionally, we find that the data provided by the lift of Ricci soliton on the lift of Ricci semi-symmetric Kenmotsu manifold that admits NSNMC in the tangent bundle are expanding. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

17 pages, 339 KiB

Open AccessArticle

Three-Dimensional Semi-Symmetric Almost α-Cosymplectic Manifolds

by Sermin Öztürk and Hakan Öztürk

Symmetry 2023, 15(11), 2022; https://doi.org/10.3390/sym15112022 - 05 Nov 2023

Viewed by 644

Abstract

The main objective of this paper is to study semi-symmetric almost

α

-cosymplectic three-manifolds. We present basic formulas for almost

α

-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost

α

-cosymplectic three-manifolds. We obtain the main [...] Read more.

The main objective of this paper is to study semi-symmetric almost

α

-cosymplectic three-manifolds. We present basic formulas for almost

α

-cosymplectic manifolds. Using curvature properties, we obtain some necessary and sufficient conditions on semi-symmetric almost

α

-cosymplectic three-manifolds. We obtain the main results under an additional condition. The paper concludes with two illustrative examples. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

13 pages, 2792 KiB

Open AccessArticle

The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space

by Serkan Çelik, Hacı Bayram Karadağ and Hatice Kuşak Samancı

Symmetry 2023, 15(6), 1191; https://doi.org/10.3390/sym15061191 - 02 Jun 2023

Cited by 1 | Viewed by 955

Abstract

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives [...] Read more.

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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13 pages, 458 KiB

Open AccessArticle

On Disks Enclosed by Smooth Jordan Curves

by José Ayala Hoffmann and Victor Ayala

Symmetry 2023, 15(6), 1140; https://doi.org/10.3390/sym15061140 - 24 May 2023

Viewed by 687

Abstract

Given a smooth-plane Jordan curve with bounded absolute curvature

κ > 0

, we determine equivalence classes of distinctive disks of radius

1 / κ

included in both plane regions separated by the curve. The bound on absolute curvature leads to a completely [...] Read more.

Given a smooth-plane Jordan curve with bounded absolute curvature

κ > 0

, we determine equivalence classes of distinctive disks of radius

1 / κ

included in both plane regions separated by the curve. The bound on absolute curvature leads to a completely symmetric trajectory behaviour with respect to the curve turning. These lead to a decomposition of the plane into a finite number of maximal regions with respect to set inclusion leading to natural lower bounds for the length an area enclosed by the curve. We present a “half version” of the Pestov–Ionin theorem, and subsequently a generalisation of the classical Blaschke rolling disk theorem. An interesting consequence is that we describe geometric conditions relying exclusively on curvature and independent of any kind of convexity that allows us to give necessary and sufficient conditions for the existence of families of rolling disks for planar domains that are not necessarily convex. We expect this approach would lead to further generalisations as, for example, characterising volumetric objects in closed surfaces as first studied by Lagunov. Although this is a classical problem in differential geometry, recent developments in industrial manufacturing when cutting along some prescribed shapes on prescribed materials have revived the necessity of a deeper understanding on disks enclosed by sufficiently smooth Jordan curves. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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12 pages, 15985 KiB

Open AccessArticle

Surfaces with Constant Negative Curvature

by Semra Kaya Nurkan and İbrahim Gürgil

Symmetry 2023, 15(5), 997; https://doi.org/10.3390/sym15050997 - 28 Apr 2023

Viewed by 1098

Abstract

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic

I I

-flat, isotropic minimal and isotropic

I I

-minimal, the constant second Gaussian curvature, [...] Read more.

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic

I I

-flat, isotropic minimal and isotropic

I I

-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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14 pages, 328 KiB

Open AccessArticle

Ricci Soliton of

CR

-Warped Product Manifolds and Their Classifications

by Yanlin Li, Sachin Kumar Srivastava, Fatemah Mofarreh, Anuj Kumar and Akram Ali

Symmetry 2023, 15(5), 976; https://doi.org/10.3390/sym15050976 - 25 Apr 2023

Cited by 26 | Viewed by 1435

Abstract

In this article, we derived an equality for

CR

-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a

CR

-warped product [...] Read more.

In this article, we derived an equality for

CR

-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a

CR

-warped product submanifolds in K

\ddot{a}

hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of

CR

-warped product submanifolds in the K

\ddot{a}

hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

18 pages, 773 KiB

Open AccessArticle

Algebraic Schouten Solitons of Three-Dimensional Lorentzian Lie Groups

by Siyao Liu

Symmetry 2023, 15(4), 866; https://doi.org/10.3390/sym15040866 - 05 Apr 2023

Viewed by 735

Abstract

In 2016, Wears defined and studied algebraic T-solitons. In this paper, we define algebraic Schouten solitons as a special T-soliton and classify the algebraic Schouten solitons associated with Levi-Civita connections, canonical connections, and Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups that have some [...] Read more.

In 2016, Wears defined and studied algebraic T-solitons. In this paper, we define algebraic Schouten solitons as a special T-soliton and classify the algebraic Schouten solitons associated with Levi-Civita connections, canonical connections, and Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups that have some product structure. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

17 pages, 834 KiB

Open AccessArticle

Differentiating the State Evaluation Map from Matrices to Functions on Projective Space

by Ghaliah Alhamzi and Edwin Beggs

Symmetry 2023, 15(2), 474; https://doi.org/10.3390/sym15020474 - 10 Feb 2023

Cited by 1 | Viewed by 722

Abstract

The pure state evaluation map from

M_{n} (C)

to

C (C P^{n - 1})

is a completely positive map of

C^{*}

-algebras intertwining the

U_{n}

symmetries on the two algebras. We show that it extends [...] Read more.

The pure state evaluation map from

M_{n} (C)

to

C (C P^{n - 1})

is a completely positive map of

C^{*}

-algebras intertwining the

U_{n}

symmetries on the two algebras. We show that it extends to a cochain map from the universal calculus on

M_{n} (C)

to the holomorphic

\bar{\partial}

calculus on

C P^{n - 1}

. The method uses connections on Hilbert

C^{*}

-bimodules. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

22 pages, 709 KiB

Open AccessArticle

The Invariants of Dual Parallel Equidistant Ruled Surfaces

by Sümeyye Gür Mazlum, Süleyman Şenyurt and Luca Grilli

Symmetry 2023, 15(1), 206; https://doi.org/10.3390/sym15010206 - 10 Jan 2023

Cited by 11 | Viewed by 1062

Abstract

In this paper, we calculate the Gaussian curvatures of the dual spherical indicatrix curves formed on unit dual sphere by the Blaschke vectors and dual instantaneous Pfaff vectors of dual parallel equidistant ruled surfaces (DPERS) and we give the relationships between these curvatures. [...] Read more.

In this paper, we calculate the Gaussian curvatures of the dual spherical indicatrix curves formed on unit dual sphere by the Blaschke vectors and dual instantaneous Pfaff vectors of dual parallel equidistant ruled surfaces (DPERS) and we give the relationships between these curvatures. In addition to—in cases where the base curves of these DPERS are closed—computing the dual integral invariants of the indicatrix curves. Additionally, we show the relationships between them. Finally, we provide an example for each of these indicatrix curves. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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33 pages, 2064 KiB

Open AccessArticle

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

by Elena Campo-Montalvo, Marián Fernández de Sevilla, J. Rafael Magdalena Benedicto and Sonia Pérez-Díaz

Symmetry 2023, 15(1), 69; https://doi.org/10.3390/sym15010069 - 26 Dec 2022

Cited by 2 | Viewed by 1379

Abstract

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve,

C

, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied [...] Read more.

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve,

C

, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them. Full article

(This article belongs to the Special Issue Symmetry/Asymmetry: Differential Geometry and Its Applications)

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