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28 pages, 1585 KB

Open AccessArticle

Higher-Dimensional Geometry and Singularity Structure of Osculating Type-II Ruled Surfaces in Lorentzian Spaces

by Mohammed Messaoudi, Marin Marin, Nidal E. Taha, Ghozail Sh. Al-Mutairi and Sayed Saber

Mathematics 2026, 14(2), 263; https://doi.org/10.3390/math14020263 - 9 Jan 2026

Cited by 1 | Viewed by 803

In Minkowski 3-space, we establish a geometric framework to osculate Type-II ruled surfaces by utilizing the Type-II Bishop frame in

(E_{1}^{3})

. Our analysis extends to higher-order singularities such as butterflies and pyramids, including explicit singularity loci. We also [...] Read more.

In Minkowski 3-space, we establish a geometric framework to osculate Type-II ruled surfaces by utilizing the Type-II Bishop frame in

(E_{1}^{3})

. Our analysis extends to higher-order singularities such as butterflies and pyramids, including explicit singularity loci. We also compare Type-II Bishop frames with rotation-minimizing frames using timelike base curves and spacelike normals. With RK4 integration, we develop a robust computational model for Weingarten surfaces and subclasses with constant curvature. The theoretical foundation for Type-II Bishop frames is extended to higher-dimensional Minkowski spaces

E_{1}^{n}

for

n > 3

through generalized Frenet-type equations and curvature functions. We determine exact stability conditions under perturbations of Bishop curvature using advanced singularity theory. The numerical implementations of our methods, including geometric modeling and relativistic geometry, demonstrate their effectiveness in both theoretical and applied contexts. Full article

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6 pages, 177 KB

Open AccessEditorial

Differentiable Manifolds and Geometric Structures

by Adara M. Blaga

Mathematics 2025, 13(7), 1082; https://doi.org/10.3390/math13071082 - 26 Mar 2025

Cited by 3 | Viewed by 1074

Abstract

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the [...] Read more.

This editorial presents 26 research articles published in the Special Issue entitled Differentiable Manifolds and Geometric Structures of the MDPI Mathematics journal, which covers a wide range of topics particularly from the geometry of (pseudo-)Riemannian manifolds and their submanifolds, providing some of the latest achievements in different areas of differential geometry, among which is counted: the geometry of differentiable manifolds with curvature restrictions such as Golden space forms, Sasakian space forms; diffeological and affine connection spaces; Weingarten and Delaunay surfaces; Chen-type inequalities for submanifolds; statistical submersions; manifolds endowed with different geometric structures (Sasakian, weak nearly Sasakian, weak nearly cosymplectic, LP-Kenmotsu, paraquaternionic); solitons (almost Ricci solitons, almost Ricci–Bourguignon solitons, gradient r-almost Newton–Ricci–Yamabe solitons, statistical solitons, solitons with semi-symmetric connections); vector fields (projective, conformal, Killing, 2-Killing) [...] Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

15 pages, 545 KB

Open AccessArticle

Modified Sweeping Surfaces in Euclidean 3-Space

by Yanlin Li, Kemal Eren, Soley Ersoy and Ana Savić

Axioms 2024, 13(11), 800; https://doi.org/10.3390/axioms13110800 - 18 Nov 2024

Cited by 18 | Viewed by 1949

Abstract

In this study, we explore the sweeping surfaces in Euclidean 3-space, utilizing the modified orthogonal frames with non-zero curvature and torsion, which allows us to consider the spine curves even if their second differentiations vanish. If the curvature of the spine curve of [...] Read more.

In this study, we explore the sweeping surfaces in Euclidean 3-space, utilizing the modified orthogonal frames with non-zero curvature and torsion, which allows us to consider the spine curves even if their second differentiations vanish. If the curvature of the spine curve of a sweeping surface has discrete zero points, the Frenet frame might undergo a discontinuous change in orientation. Therefore, the conventional parametrization with the Frenet frame of such a surface cannot be given. Thus, we introduce two types of modified sweeping surfaces by considering two types of spine curves; the first one’s curvature is not identically zero and the second one’s torsion is not identically zero. Then, we determine the criteria for classifying the coordinate curves of these two types of modified sweeping surfaces as geodesic, asymptotic, or curvature lines. Additionally, we delve into determining criteria for the modified sweeping surfaces to be minimal, developable, or Weingarten. Through our analysis, we aim to clarify the characteristics defining these surfaces. We present graphical representations of sample modified sweeping surfaces to enhance understanding and provide concrete examples that showcase their properties. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics)

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22 pages, 408 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Some Weingarten Surfaces in the Special Linear Group SL(2,

R

)

by Marian Ioan Munteanu

Mathematics 2023, 11(22), 4636; https://doi.org/10.3390/math11224636 - 13 Nov 2023

Cited by 2 | Viewed by 4332

Abstract

We classify Weingarten conoids in the real special linear group

SL (2, R)

. In particular, there is no linear Weingarten nontrivial conoids in

SL (2, R)

. We also prove that the only conoids in [...] Read more.

We classify Weingarten conoids in the real special linear group

SL (2, R)

. In particular, there is no linear Weingarten nontrivial conoids in

SL (2, R)

. We also prove that the only conoids in

SL (2, R)

with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup

N

is a Weingarten surface. Full article

(This article belongs to the Special Issue Differentiable Manifolds and Geometric Structures)

8 pages, 258 KB

Open AccessArticle

Spheres and Tori as Elliptic Linear Weingarten Surfaces

by Dong-Soo Kim, Young Ho Kim and Jinhua Qian

Mathematics 2022, 10(21), 4065; https://doi.org/10.3390/math10214065 - 1 Nov 2022

Cited by 2 | Viewed by 1746

Abstract

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres [...] Read more.

The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres and the tori immersed in the 3-dimensional unit sphere, along with the Laplace operator, the spherical Gauss map and the Gauss map associated with the elliptic linear Weingarten metric. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

19 pages, 1030 KB

Open AccessArticle

A New Approach to Rotational Weingarten Surfaces

by Paula Carretero and Ildefonso Castro

Mathematics 2022, 10(4), 578; https://doi.org/10.3390/math10040578 - 12 Feb 2022

Cited by 8 | Viewed by 4502

Abstract

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new [...] Read more.

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones. Full article

(This article belongs to the Special Issue Differential Geometry: Theory and Applications Part II)

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11 pages, 277 KB

Open AccessArticle

Characterization of Clifford Torus in Three-Spheres

by Dong-Soo Kim, Young Ho Kim and Jinhua Qian

Mathematics 2020, 8(5), 718; https://doi.org/10.3390/math8050718 - 3 May 2020

Cited by 3 | Viewed by 4461

Abstract

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the [...] Read more.

We characterize spheres and the tori, the product of the two plane circles immersed in the three-dimensional unit sphere, which are associated with the Laplace operator and the Gauss map defined by the elliptic linear Weingarten metric defined on closed surfaces in the three-dimensional sphere. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

15 pages, 786 KB

Open AccessArticle

Geometric Characterizations of Canal Surfaces in Minkowski 3-Space II

by Jinhua Qian, Mengfei Su, Xueshan Fu and Seoung Dal Jung

Mathematics 2019, 7(8), 703; https://doi.org/10.3390/math7080703 - 5 Aug 2019

Cited by 8 | Viewed by 3763

Abstract

Canal surfaces are defined and divided into nine types in Minkowski 3-space

E_{1}^{3}

, which are obtained as the envelope of a family of pseudospheres

S_{1}^{2}

, pseudohyperbolic spheres

H_{0}^{2}

, or lightlike cones

Q^{2}

, [...] Read more.

Canal surfaces are defined and divided into nine types in Minkowski 3-space

E_{1}^{3}

, which are obtained as the envelope of a family of pseudospheres

S_{1}^{2}

, pseudohyperbolic spheres

H_{0}^{2}

, or lightlike cones

Q^{2}

, whose centers lie on a space curve (resp. spacelike curve, timelike curve, or null curve). This paper focuses on canal surfaces foliated by pseudohyperbolic spheres

H_{0}^{2}

along three kinds of space curves in

E_{1}^{3}

. The geometric properties of such surfaces are presented by classifying the linear Weingarten canal surfaces, especially the relationship between the Gaussian curvature and the mean curvature of canal surfaces. Last but not least, two examples are shown to illustrate the construction of such surfaces. Full article

(This article belongs to the Special Issue Advances in singularities throughout Mathematics: Algebra, Topology, Geometry, Analysis, Number Theory, Combinatorics and Cryptography)

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7 pages, 314 KB

Open AccessArticle

Linear Weingarten Helicoidal Surfaces in Isotropic Space

by Dae Won Yoon and Jae Won Lee

Symmetry 2016, 8(11), 126; https://doi.org/10.3390/sym8110126 - 14 Nov 2016

Cited by 16 | Viewed by 5513

Abstract

In the present paper, we study helicoidal surfaces in the three-dimensional isotropic space

I^{3}

and construct helicoidal surfaces satisfying a linear equation in terms of the Gaussian curvature and the mean curvature of the surface. [...] Read more.

In the present paper, we study helicoidal surfaces in the three-dimensional isotropic space

I^{3}

and construct helicoidal surfaces satisfying a linear equation in terms of the Gaussian curvature and the mean curvature of the surface. Full article

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