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27 pages, 1140 KiB

Open AccessArticle

Singularity Analysis of Lightlike Hypersurfaces Generated by Null Cartan Curves in Minkowski Spacetime

by Xiaoming Fan, Yongsheng Zhu and Haijing Pan

Axioms 2025, 14(4), 279; https://doi.org/10.3390/axioms14040279 - 7 Apr 2025

Viewed by 153

This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface

L H_{β}

, a critical lightlike surface

L S_{β}

, and a degenerate [...] Read more.

This study investigates the singularity structures of lightlike hypersurfaces generated by null Cartan curves in Minkowski spacetime. We construct a hierarchical geometric framework consisting of a lightlike hypersurface

L H_{β}

, a critical lightlike surface

L S_{β}

, and a degenerate curve

L C_{β}

, with dimensions decreasing from 3D to 1D. Using singularity theory, we identify a novel geometric invariant

σ (t)

that governs the emergence of specific singularity types, including

C (2, 3) \times R^{2}

,

S W \times R

,

B F

,

C (B F)

,

C (2, 3, 4) \times R

, and

(2, 3, 4, 5)

-cusp. These singularities exhibit increasing degeneracy as the hierarchy progresses, with contact orders between the lightlike hyperplane

H S_{t_{0}}^{L}

and the curve

β

systematically intensifying. An explicit example demonstrates the construction of these objects and validates the theoretical results. This work establishes a systematic connection between null Cartan curves, stratified singularities, and contact geometry. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

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13 pages, 259 KiB

Open AccessArticle

Classification Results of f-Biharmonic Immersion in T-Space Forms

by Md Aquib, Mohd Iqbal and Sarvesh Kumar Yadav

Axioms 2025, 14(4), 242; https://doi.org/10.3390/axioms14040242 - 22 Mar 2025

Viewed by 179

Abstract

We investigate f-biharmonic submanifolds in T-space form, where we analyze different scenarios and provide necessary and sufficient conditions for f-biharmonicity. We also derive a non-existence result for f-biharmonic submanifolds where

ξ

and

ϕ Ω

are tangents. Finally, we derive the Chen–Ricci inequality for [...] Read more.

We investigate f-biharmonic submanifolds in T-space form, where we analyze different scenarios and provide necessary and sufficient conditions for f-biharmonicity. We also derive a non-existence result for f-biharmonic submanifolds where

ξ

and

ϕ Ω

are tangents. Finally, we derive the Chen–Ricci inequality for submanifolds of T-space forms and provide the conditions under which this inequality becomes equality. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

13 pages, 257 KiB

Open AccessArticle

Trivial Homology Groups of Warped Product Semi-Slant Submanifolds in Kenmotsu Space Forms

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

Axioms 2025, 14(3), 210; https://doi.org/10.3390/axioms14030210 - 13 Mar 2025

Viewed by 356

Abstract

This paper investigates the relationship between homology groups and warped product semi-slant submanifolds in Kenmotsu space forms. Some rigidity theorems for vanishing homology groups on warped product semi-slant submanifolds are obtained using the moving-frame method and the second fundamental form inequality. Our results [...] Read more.

This paper investigates the relationship between homology groups and warped product semi-slant submanifolds in Kenmotsu space forms. Some rigidity theorems for vanishing homology groups on warped product semi-slant submanifolds are obtained using the moving-frame method and the second fundamental form inequality. Our results are an extension of previous studies in this direction. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

36 pages, 362 KiB

Open AccessArticle

The Differential Geometry of a Space Curve via a Constant Vector in ℝ³

by Azeb Alghanemi, Ghadah Matar and Amani Saloom

Axioms 2025, 14(3), 190; https://doi.org/10.3390/axioms14030190 - 4 Mar 2025

Viewed by 494

Abstract

The differential geometry of space curves is a fascinating area of research for mathematicians and physicists, and this refers to its crucial applications in many areas. In this paper, a new method is derived to study the differential geometry of space curves. More [...] Read more.

The differential geometry of space curves is a fascinating area of research for mathematicians and physicists, and this refers to its crucial applications in many areas. In this paper, a new method is derived to study the differential geometry of space curves. More specifically, the position vector of a constant vector in

R^{3}

is given in the Frenet apparatus of a space curve, and it is implemented to study the differential geometry of the given space curve. Easy and neat proofs of various well-known results are given using this new method. Also, new results and the properties of space curves are obtained in light of this new method. More specifically, the position vectors of helices are given in simple forms. Moreover, a new frame associated with a smooth curve is obtained, as well as new curvatures associated with the new frame. The new frame and its curvatures are investigated and used to give the position vector of slant helix in a simple and memorable form. Furthermore, some non-trivial examples are given to illustrate some of the results obtained in this article. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

28 pages, 399 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Work of Cartan and Münzner on Isoparametric Hypersurfaces

by Thomas E. Cecil and Patrick J. Ryan

Axioms 2025, 14(1), 56; https://doi.org/10.3390/axioms14010056 - 13 Jan 2025

Viewed by 551

Abstract

A hypersurface

M^{n}

in a real space form

R^{n + 1}

,

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan [...] Read more.

A hypersurface

M^{n}

in a real space form

R^{n + 1}

,

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938–1940 and two papers of Münzner that were published in preprint form in the early 1970s and as journal articles in 1980–1981. These papers of Cartan and Münzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

15 pages, 275 KiB

Open AccessArticle

DDVV Inequality on Submanifolds Coupled with a Slant Factor in Quaternionic Kaehler Manifolds

by Rawan Bossly, Majid Ali Choudhary, Mohd Danish Siddiqi, Oḡuzhan Bahadır and Mehmet Gülbahar

Axioms 2025, 14(1), 6; https://doi.org/10.3390/axioms14010006 - 26 Dec 2024

Viewed by 730

Abstract

This work aims to provide generalized Wintgen inequalities for slant submanifolds embedded in quaternionic space forms, taking into consideration both semi-symmetric metric and semi-symmetric non-metric connections. Moreover, we discuss the same inequality for totally real, anti-invariant, and invariant submanifolds on quaternionic space forms, [...] Read more.

This work aims to provide generalized Wintgen inequalities for slant submanifolds embedded in quaternionic space forms, taking into consideration both semi-symmetric metric and semi-symmetric non-metric connections. Moreover, we discuss the same inequality for totally real, anti-invariant, and invariant submanifolds on quaternionic space forms, endowed with both semi-symmetric metric and semi-symmetric non-metric connections. We also characterized the equality case through specific forms of shape operators of Wintgen inequalities for these classes of submanifolds in quaternionic space forms, admitting a semi-symmetric metric connection and a semi-symmetric non-metric connection. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

14 pages, 300 KiB

Open AccessArticle

On Warped Product Pointwise Pseudo-Slant Submanifolds of LCK-Manifolds and Their Applications

by Fatimah Alghamdi

Axioms 2024, 13(11), 807; https://doi.org/10.3390/axioms13110807 - 20 Nov 2024

Viewed by 709

Abstract

The concept of pointwise slant submanifolds of a Kähler manifold was presented by Chen and Garay. This research extends this notion to a more general setting, specifically in a locally conformal Kähler manifold. We study the pointwise pseudo-slant warped products of the form [...] Read more.

The concept of pointwise slant submanifolds of a Kähler manifold was presented by Chen and Garay. This research extends this notion to a more general setting, specifically in a locally conformal Kähler manifold. We study the pointwise pseudo-slant warped products of the form

Σ^{θ} \times_{f} Σ^{⊥}

in a locally conformal Kähler manifold. Using the concept of pointwise pseudo-slant, we establish the necessary and sufficient condition for it to be characterized as a warped product submanifold. In addition, we derive several results on pointwise pseudo-slant warped products that expand previously proven main ones. Further, some examples of such submanifolds and their warped products are also given. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

10 pages, 2308 KiB

Open AccessArticle

Exact Solutions to Fractional Schrödinger–Hirota Equation Using Auxiliary Equation Method

by Guangyuan Tian and Xianji Meng

Axioms 2024, 13(10), 663; https://doi.org/10.3390/axioms13100663 - 26 Sep 2024

Cited by 1 | Viewed by 989

Abstract

In this paper, we consider the fractional Schrödinger–Hirota (FSH) equation in the sense of a conformable fractional derivative. Through a traveling wave transformation, we change the FSH equation to an ordinary differential equation. We obtain several exact solutions through the auxiliary equation method, [...] Read more.

In this paper, we consider the fractional Schrödinger–Hirota (FSH) equation in the sense of a conformable fractional derivative. Through a traveling wave transformation, we change the FSH equation to an ordinary differential equation. We obtain several exact solutions through the auxiliary equation method, including soliton, exponential and periodic solutions, which are useful to analyze the behaviors of the FSH equation. We show that the auxiliary equation method improves the speed of the discovery of exact solutions. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

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