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12 pages, 310 KB

Open AccessArticle

Classification of Three-Dimensional Contact Metric Manifolds with Almost-Generalized

Ƶ

-Solitons

by Shahroud Azami, Mehdi Jafari and Daniele Ettore Otera

Mathematics 2025, 13(23), 3765; https://doi.org/10.3390/math13233765 - 24 Nov 2025

Viewed by 224

Abstract

This work investigates the classification of three-dimensional complete contact metric manifolds that are non-Sasakian and satisfy the relation

Q ξ = σ ξ

, focusing on those that support an almost-generalized

Ƶ

-soliton. In the scenario where

σ

is constant, we prove that [...] Read more.

This work investigates the classification of three-dimensional complete contact metric manifolds that are non-Sasakian and satisfy the relation

Q ξ = σ ξ

, focusing on those that support an almost-generalized

Ƶ

-soliton. In the scenario where

σ

is constant, we prove that if a generalized

Ƶ

-soliton

(M^{n}, g, δ, η, V, μ, Λ)

satisfies the condition

g (V, ξ) = 0

, then

M^{n}

must be either an Einstein manifold or locally isometric to the Lie group

E (1, 1)

. Comparable classifications are obtained for

(κ, μ, ϑ)

-contact metric manifolds. Furthermore, we explore situations in which the potential vector field aligns with the Reeb vector field. We then provide the corresponding structural characterizations. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

16 pages, 678 KB

Open AccessArticle

Lorentzian Structure and Curvature Analysis of Osculating Type-2 Ruled Surfaces via the Type-2 Bishop Frame

by Mohammed Messaoudi, Emad Solouma, Mohammed N. Alshehri, Abdulrahman F. Aljohani and Marin Marin

Mathematics 2025, 13(21), 3464; https://doi.org/10.3390/math13213464 - 30 Oct 2025

Viewed by 367

Abstract

This study investigates the geometry of osculating type-2 ruled surfaces in Minkowski 3-space

E_{1}^{3}

, formulated through the Type-2 Bishop frame associated with a spacelike curve whose principal normal is timelike and binormal is spacelike. Using the hyperbolic transformation linking the [...] Read more.

This study investigates the geometry of osculating type-2 ruled surfaces in Minkowski 3-space

E_{1}^{3}

, formulated through the Type-2 Bishop frame associated with a spacelike curve whose principal normal is timelike and binormal is spacelike. Using the hyperbolic transformation linking the Frenet–Serret and Bishop frames, we analyze how the Bishop curvatures

ζ_{1}

and

ζ_{2}

affect the geometric behavior and formation of such surfaces. Explicit criteria are derived for cylindrical, developable, and minimal configurations, together with analytical expressions for Gaussian and mean curvatures. We also determine the conditions under which the base curve behaves as a geodesic, asymptotic line, or line of curvature. Several illustrative examples in Minkowski 3-space are provided to visualize the geometric influence of

ζ_{1}

and

ζ_{2}

on flatness, minimality, and developability. Overall, the Type-2 Bishop frame offers a smooth and effective framework for characterizing Lorentzian geometry and symmetry of osculating ruled surfaces, extending classical Euclidean results to the Minkowski setting. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

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16 pages, 722 KB

Open AccessArticle

Geometric Invariants and Evolution of RM Hasimoto Surfaces in Minkowski 3-Space

E_{1}^{3}

by Emad Solouma, Sayed Saber, Marin Marin and Haci Mehmet Baskonus

Mathematics 2025, 13(21), 3420; https://doi.org/10.3390/math13213420 - 27 Oct 2025

Viewed by 295

Abstract

Research on surfaces generated by curves plays a central role in linking differential geometry with physical applications, especially following Hasimoto’s transformation and the development of Hasimoto-inspired surface models. In this work, we introduce a new class of such surfaces, referred to as RM [...] Read more.

Research on surfaces generated by curves plays a central role in linking differential geometry with physical applications, especially following Hasimoto’s transformation and the development of Hasimoto-inspired surface models. In this work, we introduce a new class of such surfaces, referred to as RM Hasimoto surfaces, constructed by employing the rotation-minimizing (RM) Darboux frame along both timelike and spacelike curves in Minkowski 3-space

E_{1}^{3}

. In contrast to the classical Hasimoto surfaces defined via the Frenet or standard Darboux frames, the RM approach eliminates torsional difficulties and reduces redundant rotational effects. This leads to more straightforward expressions for the first and second fundamental forms, as well as for the Gaussian and mean curvatures, and facilitates a clear classification of key parameter curves. Furthermore, we establish the associated evolution equations, analyze the resulting geometric invariants, and present explicit examples based on timelike and spacelike generating curves. The findings show that adopting the RM Darboux frame provides greater transparency in Lorentzian surface geometry, yielding sharper characterizations and offering new perspectives on relativistic vortex filaments, magnetic field structures, and soliton behavior. Thus, the RM framework opens a promising direction for both theoretical studies and practical applications of surface geometry in Minkowski space. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

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11 pages, 265 KB

Open AccessFeature PaperArticle

Pair of Associated η-Ricci–Bourguignon Almost Solitons with Vertical Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds

by Mancho Manev

Mathematics 2025, 13(11), 1863; https://doi.org/10.3390/math13111863 - 3 Jun 2025

Cited by 1 | Viewed by 678

Abstract

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as [...] Read more.

The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an

η

-Ricci–Bourguignon almost soliton, where

η

is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated

η

-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

12 pages, 253 KB

Open AccessArticle

Pinching Results on Totally Real Submanifolds of a Locally Conformal Kähler Manifolds

by Noura M. Alhouiti, Ali H. Alkhaldi, Akram Ali and Piscoran Laurian-Ioan

Mathematics 2025, 13(10), 1682; https://doi.org/10.3390/math13101682 - 21 May 2025

Viewed by 557

Abstract

This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. [...] Read more.

This paper investigates the relationship between pseudo-umbilical and minimal totally real submanifolds in locally conformal Kähler space forms. Some rigidity theorems and an integral inequality are obtained using the moving-frame method and the DDVV inequality. Our results extend this line of previous research. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

15 pages, 280 KB

Open AccessArticle

Integral Formulae and Applications for Compact Riemannian Hypersurfaces in Riemannian and Lorentzian Manifolds Admitting Concircular Vector Fields

by Mona Bin-Asfour, Kholoud Saad Albalawi and Mohammed Guediri

Mathematics 2025, 13(10), 1672; https://doi.org/10.3390/math13101672 - 20 May 2025

Viewed by 638

Abstract

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined [...] Read more.

This paper investigates compact Riemannian hypersurfaces immersed in

(n + 1)

-dimensional Riemannian or Lorentzian manifolds that admit concircular vector fields, also known as closed conformal vector fields (CCVFs). We focus on the support function of the hypersurface, which is defined as the component of the conformal vector field along the unit-normal vector field, and derive an expression for its Laplacian. Using this, we establish integral formulae for hypersurfaces admitting CCVFs. These results are then extended to compact Riemannian hypersurfaces isometrically immersed in Riemannian or Lorentzian manifolds with constant sectional curvatures, highlighting the crucial role of CCVFs in the study of hypersurfaces. We apply these results to provide characterizations of compact Riemannian hypersurfaces in Euclidean space

R^{n + 1}

, Euclidean sphere

S^{n + 1}

, and de Sitter space

S_{1}^{n + 1}

. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

33 pages, 458 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Recent Developments in Chen’s Biharmonic Conjecture and Some Related Topics

by Bang-Yen Chen

Mathematics 2025, 13(9), 1417; https://doi.org/10.3390/math13091417 - 25 Apr 2025

Viewed by 1515

Abstract

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the [...] Read more.

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e.,

Δ^{2} x = 0

. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

Review

Jump to: Research

45 pages, 567 KB

Open AccessReview

Magnetic Curves in Differential Geometry: A Comprehensive Survey

by Bang-Yen Chen, Foued Aloui, Md Ajmal Khan and Majid Ali Choudhary

Mathematics 2025, 13(23), 3849; https://doi.org/10.3390/math13233849 - 1 Dec 2025

Viewed by 326

Abstract

The concept of “magnetic lines of force,” or “magnetic curves”, was introduced in the 1830s by Michael Faraday (1791–1867); his work provided the foundation for the modern understanding of magnetic fields. In differential geometry, a magnetic curve is a concept that arises from [...] Read more.

The concept of “magnetic lines of force,” or “magnetic curves”, was introduced in the 1830s by Michael Faraday (1791–1867); his work provided the foundation for the modern understanding of magnetic fields. In differential geometry, a magnetic curve is a concept that arises from the intersection of geometry and physics. These curves represent the trajectories of a charged particle experiencing the Lorentz force as it travels through a magnetic field. These curves have garnered significant interest due to their intricate geometric properties and diverse applications. This paper provides a comprehensive exploration of magnetic curves, delving into their fundamental characteristics and classification. Full article

(This article belongs to the Special Issue Analysis on Differentiable Manifolds)

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