Symmetry

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11 pages, 251 KB

Open AccessArticle

On Finite-Type Gauss Maps of Quadric Surfaces in Euclidean 3-Space

by Mutaz Al-Sabbagh

Symmetry 2026, 18(5), 755; https://doi.org/10.3390/sym18050755 - 28 Apr 2026

Viewed by 228

This paper focuses on the study of quadric surfaces in three-dimensional Euclidean space that satisfy the finite II-type Gauss map condition, a concept introduced firstly by B.-Y. Chen in its first fundamental form; later it was applied by other researchers in the second [...] Read more.

This paper focuses on the study of quadric surfaces in three-dimensional Euclidean space that satisfy the finite II-type Gauss map condition, a concept introduced firstly by B.-Y. Chen in its first fundamental form; later it was applied by other researchers in the second and third fundamental forms. This study’s primary finding is that spheres are the only quadric surfaces with this characteristic. This suggests a particular and significant grouping in the more general class of quadric surfaces according to their finite-type properties with respect to the second fundamental form. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

12 pages, 4390 KB

Open AccessArticle

Dual THA-Surfaces in 3-Dimensional Galilean Space

by Yanlin Li, Mehraj Ahmad Lone and Arshi Yousuf

Symmetry 2026, 18(3), 409; https://doi.org/10.3390/sym18030409 - 26 Feb 2026

Cited by 2 | Viewed by 326

Abstract

The Galilean space

G^{3}

occupies a central position in the analysis of classical kinematics and provides the geometric framework for representing Newtonian space-time. In this paper, we concentrate on classifying dual THA-surfaces in

G^{3}

and classify specific types of these surfaces [...] Read more.

The Galilean space

G^{3}

occupies a central position in the analysis of classical kinematics and provides the geometric framework for representing Newtonian space-time. In this paper, we concentrate on classifying dual THA-surfaces in

G^{3}

and classify specific types of these surfaces under geometric conditions, including zero dual Gaussian curvature and zero dual mean curvature. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

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11 pages, 264 KB

Open AccessArticle

Maximal Elements of Preferences on Compact Spaces from Optimization of One-Way Utilities

by Gianni Bosi, Gabriele Sbaiz and Magalì Zuanon

Symmetry 2026, 18(1), 155; https://doi.org/10.3390/sym18010155 - 14 Jan 2026

Viewed by 279

Abstract

The search for maximal elements of preference relations has been recently related to the optimization of one-way utilities on compact topological spaces. In this paper, we deepen this study by referring to upper semicontinuous finite Richter–Peleg multi-utility representations of preorders. We provide necessary [...] Read more.

The search for maximal elements of preference relations has been recently related to the optimization of one-way utilities on compact topological spaces. In this paper, we deepen this study by referring to upper semicontinuous finite Richter–Peleg multi-utility representations of preorders. We provide necessary and sufficient conditions for the existence of representations of this kind and then, under the assumption of near-completeness, we characterize the identification of all maximal elements by maximizing all functions in an appropriate representation under compactness. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

19 pages, 1365 KB

Open AccessArticle

Parallel Darboux Equidistant Ruled Surfaces in E³

by Ceyda Cevahir Yıldız, Süleyman Şenyurt and Luca Grilli

Symmetry 2026, 18(1), 111; https://doi.org/10.3390/sym18010111 - 7 Jan 2026

Viewed by 484

Abstract

In this study, equidistant ruled surfaces generated by the Darboux vector, which has significant kinematic importance and characterizes the instantaneous rotation of a moving frame, are investigated specifically for the Frenet frame. By establishing a structural relationship between a surface and its equidistant [...] Read more.

In this study, equidistant ruled surfaces generated by the Darboux vector, which has significant kinematic importance and characterizes the instantaneous rotation of a moving frame, are investigated specifically for the Frenet frame. By establishing a structural relationship between a surface and its equidistant ruled surface, transition formulas are provided for shape operators, Gaussian and mean curvatures, and fundamental forms, revealing that the equidistant surface is a scaled transformation of the original one. The obtained results demonstrate that both surfaces are developable and that the geometric properties of the equidistant ruled surfaces can be expressed dependently on each other. Furthermore, it is shown that the geometric character of the equidistant surface, including the invariance of asymptotic lines and the preservation of umbilical points under constant angle conditions, is determined by the rotational dynamics of the base curve. These findings constitute a theoretical foundation for cases involving the use of Darboux axes of different frames in higher dimensions or the investigation of similar structures in different geometric spaces. The geometric interpretation of this theoretical framework is elucidated through the fundamental properties of the surfaces. Finally, a concrete example is presented, where the symmetry of the central planes of the equidistant ruled surfaces at appropriate points is visualized using Maple 2017 software. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

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14 pages, 1498 KB

Open AccessArticle

Construction of a New Hypersurface Family Using the Spherical Product in Minkowski Geometry

by Sezgin Büyükkütük, Ilim Kişi, Günay Öztürk and Emre Kişi

Symmetry 2026, 18(1), 77; https://doi.org/10.3390/sym18010077 - 2 Jan 2026

Viewed by 414

Abstract

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space

E^{n}

, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When [...] Read more.

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space

E^{n}

, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When

(n - 1)

-planar curves are considered, this construction enables the generation of hypersurfaces in n-dimensional spaces. Building upon this geometric framework, we conduct the first-ever investigation of spherical product hypersurfaces in the context of Minkowski geometry. We define these hypersurfaces in four-dimensional Minkowski space

E_{1}^{4}

and derive explicit expressions for their Gaussian and mean curvatures. We also determine the conditions under which such hypersurfaces are flat or minimal. Furthermore, we reinterpret certain hyperquadrics as specific instances of spherical product hypersurfaces in

E_{1}^{4}

, supported by visual illustrations. Finally, we extend the construction to arbitrary-dimensional Minkowski spaces, providing a unified formulation for spherical product hypersurfaces across higher-dimensional Lorentzian geometries. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

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20 pages, 349 KB

Open AccessArticle

Some Solitons on Anti-Invariant Submanifolds of Para-Sasakian Manifold Admitting Semi-Symmetric Non-Metric Connection

by Abhijit Mandal, Mustafa Yıldırım, Adel Mohammed Ali Al-Qashbari and Mohammad Nazrul Islam Khan

Symmetry 2026, 18(1), 30; https://doi.org/10.3390/sym18010030 - 23 Dec 2025

Viewed by 801

Abstract

The objective of this study is to investigate

ρ

-Einstein solitons on submanifolds of a para-Sasakian manifold under certain curvature conditions. The novelty of this work lies in the characterization of

ρ

-Einstein solitons on anti-invariant submanifolds of a para-Sasakian manifold equipped with [...] Read more.

The objective of this study is to investigate

ρ

-Einstein solitons on submanifolds of a para-Sasakian manifold under certain curvature conditions. The novelty of this work lies in the characterization of

ρ

-Einstein solitons on anti-invariant submanifolds of a para-Sasakian manifold equipped with a semi-symmetric non-metric connection, where the structure vector field is taken as the potential vector field. We establish several significant results concerning the classification of

ρ

-Einstein solitons with respect to the

W_{3}

-curvature tensor and the semi-symmetric non-metric connection. Moreover, we construct a non-trivial example of an anti-invariant submanifold of a five-dimensional para-Sasakian manifold by solving an associated system of partial differential equations. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

18 pages, 301 KB

Open AccessArticle

Casorati-Type Inequalities for Submanifolds in S-Space Forms with Semi-Symmetric Connection

by Md Aquib

Symmetry 2025, 17(7), 1100; https://doi.org/10.3390/sym17071100 - 9 Jul 2025

Cited by 2 | Viewed by 691

Abstract

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized

δ

-Casorati curvatures [...] Read more.

The primary aim of this paper is to establish two sharp geometric inequalities concerning submanifolds of S-space forms equipped with semi-symmetric metric connections (SSMCs). Specifically, we derive new inequalities involving the generalized normalized

δ

-Casorati curvatures

δ_{c} (t; q_{1} + q_{2} - 1)

and

{\hat{δ}}_{c} (t; q_{1} + q_{2} - 1)

for bi-slant submanifolds. The cases in which equality holds are thoroughly examined, offering a deeper understanding of the geometric structure underlying such submanifolds. In addition, we present several immediate applications that highlight the relevance of our findings, and we support the article with illustrative examples. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

15 pages, 328 KB

Open AccessArticle

Null Hybrid Curves and Some Characterizations of Null Hybrid Bertrand Curves

by Jeta Alo

Symmetry 2025, 17(2), 312; https://doi.org/10.3390/sym17020312 - 19 Feb 2025

Cited by 1 | Viewed by 1078

Abstract

In this paper, we investigate null curves in

R_{2}^{4}

, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in

R_{2}^{4}

. We [...] Read more.

In this paper, we investigate null curves in

R_{2}^{4}

, the four-dimensional Minkowski space of index 2, utilizing the concept of hybrid numbers. Hybrid and spatial hybrid-valued functions of a single variable describe a curve in

R_{2}^{4}

. We first derive Frenet formulas for a null curve in

R_{2}^{3}

, the three-dimensional Minkowski space of index 2, by means of spatial hybrid numbers. Next, we apply the Frenet formulas for the associated null spatial hybrid curve corresponding to a null hybrid curve in order to derive the Frenet formulas for this curve in

R_{2}^{4}

. This approach is simpler and more efficient than the classical differential geometry methods and enables us to determine a null curve in

R_{2}^{3}

corresponding to the null curve in

R_{2}^{4}

. Additionally, we provide an example of a null hybrid curve, demonstrate the construction of its Frenet frame, and calculate the curvatures of the curve. Finally, we introduce null hybrid Bertrand curves, and by using their symmetry properties, we provide some characterizations of these curves. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

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14 pages, 276 KB

Open AccessArticle

Eigenvalues for the Generalized Laplace Operator of Slant Submanifolds in the Sasakian Space Forms Admitting Semi-Symmetric Metric Connection

by Ibrahim Al-Dayel, Meraj Ali Khan and Sudhakar Kumar Chaubey

Symmetry 2025, 17(2), 279; https://doi.org/10.3390/sym17020279 - 11 Feb 2025

Viewed by 750

Abstract

This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine [...] Read more.

This study is focused on pioneering new upper bounds on mean curvature and constant sectional curvature relative to the first positive eigenvalue of the generalized Laplacian operator in the differentiable manifolds with a semi-symmetric metric connection. Multiple approaches are being explored to determine the principal eigenvalue for the generalized-Laplacian operator in closed oriented-slant submanifolds within a Sasakian space form (ssf) with a semi-symmetric metric (ssm) connection. By utilizing our findings on the Laplacian, we extend several Reilly-type inequalities to the generalized Laplacian on slant submanifolds within a unit sphere with a semi-symmetric metric (ssm) connection. The research is concluded with a detailed examination of specific scenarios. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 4th Edition)

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