Research

13 pages, 261 KiB

Open AccessArticle

On the Conjecture over Dimensions of Associated Lie Algebra to the Isolated Singularities

by Naveed Hussain, Ahmad N. Al-Kenani and Muhammad Asif

Axioms 2024, 13(4), 216; https://doi.org/10.3390/axioms13040216 - 25 Mar 2024

Viewed by 501

Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of classification theory of isolated singularities. Any Lie algebra that [...] Read more.

Lie algebra plays an important role in the study of singularity theory and other fields of sciences. Finding numerous invariants linked with isolated singularities has always been a primary interest in the field of classification theory of isolated singularities. Any Lie algebra that characterizes simple singularity produces a natural question. The study of properties such as to find the dimensions of newly defined algebra is a remarkable work. Hussain, Yau and Zuo have found a new class of Lie algebra

L_{k} (V)

,

k \geq 1

, i.e., Der (

M_{k} (V), M_{k} (V)

) and proposed a conjecture over its dimension

δ_{k} (V)

for

k \geq 0

. Later, they proved it true for k up to

k = 1, 2, 3, 4, 5

. In this work, the main concern is whether it is true for a higher value of k. According to this, we first calculate the dimension of Lie algebra

L_{k} (V)

for

k = 6

and then compute the upper estimate conjecture of fewnomial isolated singularities. Additionally, we also justify the inequality conjecture

δ_{k + 1} (V) < δ_{k} (V)

for

k = 6

. Our calculated results are innovative and serve as a new addition to the study of singularity theory. Full article

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

11 pages, 243 KiB

Open AccessArticle

The Bifurcations of Completely Integrable Holonomic Systems of First-Order Differential Equations

by Jingbo Xu, Kangping Liu and Xiaoliang Cheng

Axioms 2024, 13(1), 55; https://doi.org/10.3390/axioms13010055 - 16 Jan 2024

Viewed by 770

Abstract

As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are

R^{+}

-simple and stable. Using this result, the parametric differential equation [...] Read more.

As an application of the Legendrian singularity theory, we classify the bifurcations of a holonomic first-order differential equation with a complete integral. The equations satisfy that the one-parameter integral diagrams are

R^{+}

-simple and stable. Using this result, the parametric differential equation models in electrical power systems and engineering can be studied. Full article

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

20 pages, 363 KiB

Open AccessArticle

Optimal Inequalities on (α,β)-Type Almost Contact Manifold with the Schouten–Van Kampen Connection

by Mohd Danish Siddiqi and Ali H. Hakami

Axioms 2023, 12(12), 1082; https://doi.org/10.3390/axioms12121082 - 27 Nov 2023

Viewed by 1078

Abstract

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or

(α, β)

-type almost contact manifolds endowed with the Schouten–Van Kampen connection (

S V K

-connection), including generalized normalized

δ

-Casorati Curvatures (

δ

-CC). [...] Read more.

In the current research, we develop optimal inequalities for submanifolds in trans-Sasakian manifolds or

(α, β)

-type almost contact manifolds endowed with the Schouten–Van Kampen connection (

S V K

-connection), including generalized normalized

δ

-Casorati Curvatures (

δ

-CC). We also discuss submanifolds on which the equality situations occur. Lastly, we provided an example derived from this research. Full article

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

15 pages, 279 KiB

Open AccessArticle

Semi-Conformally Flat Singly Warped Product Manifolds and Applications

by Samesh Shenawy, Alaa Rabie, Uday Chand De, Carlo Mantica and Nasser Bin Turki

Axioms 2023, 12(12), 1078; https://doi.org/10.3390/axioms12121078 - 24 Nov 2023

Viewed by 828

Abstract

This paper investigates singly warped product manifolds admitting semi-conformal curvature tensors. The form of the Riemann tensor and Ricci tensor of the base and fiber manifolds of a semi-conformally flat singly warped product manifold are provided. It is demonstrated that the fiber manifold [...] Read more.

This paper investigates singly warped product manifolds admitting semi-conformal curvature tensors. The form of the Riemann tensor and Ricci tensor of the base and fiber manifolds of a semi-conformally flat singly warped product manifold are provided. It is demonstrated that the fiber manifold of a semi-conformally flat warped product manifold has a constant curvature. Sufficient requirements on the warping function to ensure that the base manifold is a quasi-Einstein or an Einstein manifold are provided. Full article

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

16 pages, 621 KiB

Open AccessArticle

On the Timelike Circular Surface and Singularities in Minkowski 3-Space

by Areej A. Almoneef and Rashad A. Abdel-Baky

Axioms 2023, 12(10), 989; https://doi.org/10.3390/axioms12100989 - 19 Oct 2023

Viewed by 836

Abstract

In this paper, we have parameterized a timelike (

T l i k e

) circular surface (

CI surface

) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a [...] Read more.

In this paper, we have parameterized a timelike (

T l i k e

) circular surface (

CI surface

) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a

T l i k e

roller coaster surface (

RCO surface

) to be a flat or minimal surface is examined in detail. Further, we illustrate the approach’s outcomes with a number of pertinent examples. Full article

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

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14 pages, 287 KiB

Open AccessArticle

Inequalities for the Generalized Normalized δ-Casorati Curvatures of Submanifolds in Golden Riemannian Manifolds

by Majid Ali Choudhary and Ion Mihai

Axioms 2023, 12(10), 952; https://doi.org/10.3390/axioms12100952 - 08 Oct 2023

Viewed by 727

Abstract

In the present article, we consider submanifolds in golden Riemannian manifolds with constant golden sectional curvature. On such submanifolds, we prove geometric inequalities for the Casorati curvatures. The submanifolds meeting the equality cases are also described. Full article

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

55 pages, 531 KiB

Open AccessArticle

On the Geometry of the Null Tangent Bundle of a Pseudo-Riemannian Manifold

by Mohamed Tahar Kadaoui Abbassi, Khadija Boulagouaz and Giovanni Calvaruso

Axioms 2023, 12(10), 903; https://doi.org/10.3390/axioms12100903 - 22 Sep 2023

Viewed by 544

Abstract

When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on [...] Read more.

When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on the tangent bundle of a pseudo-Riemannian manifold and we investigate the geometry of the null tangent bundle as a lightlike hypersurface equipped with an induced g-natural metric. Full article

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

10 pages, 926 KiB

Open AccessArticle

Surface Family Pair with Bertrand Pair as Mutual Curvature Lines in Three-Dimensional Lie Group

by Awatif Al-Jedani and Rashad A. Abdel-Baky

Axioms 2023, 12(9), 830; https://doi.org/10.3390/axioms12090830 - 28 Aug 2023

Viewed by 536

Abstract

This paper is on deducing the necessary and sufficient conditions of a surface family pair with a Bertrand pair as mutual curvature lines in three-dimensional Lie group

G

. As a result, the consequence for the ruled surface family pair is also extrapolated. [...] Read more.

This paper is on deducing the necessary and sufficient conditions of a surface family pair with a Bertrand pair as mutual curvature lines in three-dimensional Lie group

G

. As a result, the consequence for the ruled surface family pair is also extrapolated. Meanwhile, examples are specified to show the surface family with common Bertrand geodesic curves. Full article

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

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15 pages, 280 KiB

Open AccessArticle

On Some Quasi-Curves in Galilean Three-Space

by Ayman Elsharkawy, Yusra Tashkandy, Walid Emam, Clemente Cesarano and Noha Elsharkawy

Axioms 2023, 12(9), 823; https://doi.org/10.3390/axioms12090823 - 27 Aug 2023

Viewed by 896

Abstract

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute [...] Read more.

In this paper, the quasi-frame and quasi-formulas are introduced in Galilean three-space. In addition, the quasi-Bertrand and the quasi-Mannheim curves are studied. It is proven that the angle between the tangents of two quasi-Bertrand or quasi-Mannhiem curves is not constant. Furthermore, the quasi-involute is studied. Moreover, we prove that there is no quasi-evolute curve in Galilean three-space. Also, we introduce quasi-Smarandache curves in Galilean three-space. Finally, we demonstrate an illustrated example to present our findings. Full article

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

11 pages, 272 KiB

Open AccessArticle

Kropina Metrics with Isotropic Scalar Curvature

by Liulin Liu, Xiaoling Zhang and Lili Zhao

Axioms 2023, 12(7), 611; https://doi.org/10.3390/axioms12070611 - 21 Jun 2023

Cited by 1 | Viewed by 769

Abstract

In this paper, we study Kropina metrics with isotropic scalar curvature. First, we obtain the expressions of Ricci curvature tensor and scalar curvature. Then, we characterize the Kropina metrics with isotropic scalar curvature on by tensor analysis. Full article

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

14 pages, 3785 KiB

Open AccessArticle

Confocal Families of Hyperbolic Conics via Quadratic Differentials

by Joel Langer and David Singer

Axioms 2023, 12(6), 507; https://doi.org/10.3390/axioms12060507 - 24 May 2023

Cited by 1 | Viewed by 957

Abstract

We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that [...] Read more.

We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics. Full article

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

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16 pages, 403 KiB

Open AccessArticle

Quaternionic Shape Operator and Rotation Matrix on Ruled Surfaces

by Yanlin Li and Abdussamet Çalışkan

Axioms 2023, 12(5), 486; https://doi.org/10.3390/axioms12050486 - 17 May 2023

Cited by 18 | Viewed by 1137

Abstract

In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, [...] Read more.

In this article, we examine the relationship between Darboux frames along parameter curves and the Darboux frame of the base curve of the ruled surface. We derive the equations of the quaternionic shape operators, which can rotate tangent vectors around the normal vector, and find the corresponding rotation matrices. Using these operators, we examine the Gauss curvature and mean curvature of the ruled surface. We explore how these properties are found by the use of Frenet vectors instead of generator vectors. We provide illustrative examples to better demonstrate the concepts and results discussed. Full article

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

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13 pages, 305 KiB

Open AccessArticle

On an Indefinite Metric on a Four-Dimensional Riemannian Manifold

by Dimitar Razpopov, Georgi Dzhelepov and Iva Dokuzova

Axioms 2023, 12(5), 432; https://doi.org/10.3390/axioms12050432 - 27 Apr 2023

Viewed by 727

Abstract

Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type

(1, 1)

, whose fourth power is minus [...] Read more.

Our research focuses on the tangent space of a point on a four-dimensional Riemannian manifold. Besides having a positive definite metric, the manifold is endowed with an additional tensor structure of type

(1, 1)

, whose fourth power is minus the identity. The additional structure is skew-circulant and compatible with the metric, such that an isometry is induced in every tangent space on the manifold. Both structures define an indefinite metric. With the help of the indefinite metric, we determine circles in different two-planes in the tangent space on the manifold. We also calculate the length and area of the circles. On a smooth closed curve, such as a circle, we define a vector force field. Further, we obtain the circulation of the vector force field along the curve, as well as the flux of the curl of this vector force field across the curve. Finally, we find a relation between these two values, which is an analog of the well-known Green’s formula in the Euclidean space. Full article

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

82 pages, 748 KiB

Open AccessArticle

C-R Immersions and Sub-Riemannian Geometry

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Axioms 2023, 12(4), 329; https://doi.org/10.3390/axioms12040329 - 28 Mar 2023

Viewed by 977

Abstract

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form

θ

, we consider natural

ϵ

-contractions, i.e., contractions

g_{ϵ}^{M}

of the Levi form

G_{θ}

, such that the norm [...] Read more.

On any strictly pseudoconvex CR manifold M, of CR dimension n, equipped with a positively oriented contact form

θ

, we consider natural

ϵ

-contractions, i.e., contractions

g_{ϵ}^{M}

of the Levi form

G_{θ}

, such that the norm of the Reeb vector field T of

(M, θ)

is of order

O (ϵ^{- 1})

. We study isopseudohermitian (i.e.,

f^{*} Θ = θ

) Cauchy–Riemann immersions

f : M \to (A, Θ)

between strictly pseudoconvex CR manifolds M and A, where

Θ

is a contact form on A. For every contraction

g_{ϵ}^{A}

of the Levi form

G_{Θ}

, we write the embedding equations for the immersion

f : M \to (A, g_{ϵ}^{A})

. A pseudohermitan version of the Gauss equation for an isopseudohermitian C-R immersion is obtained by an elementary asymptotic analysis as

ϵ \to 0^{+}

. For every isopseudohermitian immersion

f : M \to S^{2 N + 1}

into a sphere

S^{2 N + 1} \subset C^{N + 1}

, we show that Webster’s pseudohermitian scalar curvature R of

(M, θ)

satisfies the inequality

R \leq 2 n [(f^{*} g_{Θ}) (T, T) + n + 1] + \frac{1}{2} {∥ H (f) ∥_{g_{Θ}^{f}}^{2} + ∥ {trace}_{G_{θ}} Π_{H (M)} (\nabla^{⊤} - \nabla) ∥_{f^{*} g_{Θ}}^{2}}

with equality if and only if

B (f) = 0

and

\nabla^{⊤} = \nabla

on

H (M) \otimes H (M)

. This gives a pseudohermitian analog to a classical result by S-S. Chern on minimal isometric immersions into space forms. Full article

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

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