MDPI - Publisher of Open Access Journals

13 pages, 291 KiB

Open AccessArticle

Two Special Types of Curves in Lorentzian α-Sasakian 3-Manifolds

by Xiawei Chen and Haiming Liu

Symmetry 2023, 15(5), 1077; https://doi.org/10.3390/sym15051077 - 12 May 2023

Viewed by 1319

In this paper, we focus on the research and analysis of the geometric properties and symmetry of slant curves and contact magnetic curves in Lorentzian

α

-Sasakian 3-manifolds. To do this, we define the notion of Lorentzian cross product. From the perspectives of [...] Read more.

In this paper, we focus on the research and analysis of the geometric properties and symmetry of slant curves and contact magnetic curves in Lorentzian

α

-Sasakian 3-manifolds. To do this, we define the notion of Lorentzian cross product. From the perspectives of the Legendre and non-geodesic curves, we found the ratio relationship between the curvature and torsion of the slant curve and contact magnetic curve in the Lorentzian

α

-Sasakian 3-manifolds. Moreover, we utilized the property of the contact magnetic curve to characterize the manifold as Lorentzian

α

-Sasakian and to find the slant curve type of the Frenet contact magnetic curve. Furthermore, we found an example to verify the geometric properties of the slant curve and contact magnetic curve in the Lorentzian

α

-Sasakian 3-manifolds. Full article

(This article belongs to the Special Issue Symmetry and Its Application in Differential Geometry and Topology, 2nd Edition)

14 pages, 352 KiB

Open AccessArticle

Generalized Lorentzian Sasakian-Space-Forms with

M

-Projective Curvature Tensor

by D. G. Prakasha, M. R. Amruthalakshmi, Fatemah Mofarreh and Abdul Haseeb

Mathematics 2022, 10(16), 2869; https://doi.org/10.3390/math10162869 - 11 Aug 2022

Cited by 3 | Viewed by 1477

Abstract

In this note, the generalized Lorentzian Sasakian-space-form

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfying certain constraints on the

M

-projective curvature tensor

W^{*}

is considered. Here, we characterize the structure [...] Read more.

In this note, the generalized Lorentzian Sasakian-space-form

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfying certain constraints on the

M

-projective curvature tensor

W^{*}

is considered. Here, we characterize the structure

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

when it is, respectively,

M

-projectively flat,

M

-projectively semisymmetric,

M

-projectively pseudosymmetric, and

φ - M

-projectively semisymmetric. Moreover,

M_{1}^{2 n + 1} (f_{1}, f_{2}, f_{3})

satisfies the conditions

W^{*} (ζ, V_{1}) \cdot S = 0

W^{*} (ζ, V_{1}) \cdot R = 0

and

W^{*} (ζ, V_{1}) \cdot W^{*} = 0

are also examined. Finally, illustrative examples are given for obtained results. Full article

(This article belongs to the Special Issue Analytic and Geometric Inequalities: Theory and Applications)

11 pages, 775 KiB

Open AccessFeature PaperArticle

Slant Curves in Contact Lorentzian Manifolds with CR Structures

by Ji-Eun Lee

Mathematics 2020, 8(1), 46; https://doi.org/10.3390/math8010046 - 1 Jan 2020

Cited by 7 | Viewed by 2538

Abstract

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the

\hat{\nabla}

-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we [...] Read more.

In this paper, we first find the properties of the generalized Tanaka–Webster connection in a contact Lorentzian manifold. Next, we find that a necessary and sufficient condition for the

\hat{\nabla}

-geodesic is a magnetic curve (for ∇) along slant curves. Finally, we prove that when

c \leq 0

, there does not exist a non-geodesic slant Frenet curve satisfying the

\hat{\nabla}

-Jacobi equations for the

\hat{\nabla}

-geodesic vector fields in M. Thus, we construct the explicit parametric equations of pseudo-Hermitian pseudo-helices in Lorentzian space forms

M_{1}^{3} (\hat{H})

for

\hat{H} = 2 c > 0

. Full article

(This article belongs to the Special Issue Sasakian Space)

13 pages, 276 KiB

Open AccessArticle

Slant Curves and Contact Magnetic Curves in Sasakian Lorentzian 3-Manifolds

by Ji-Eun Lee

Symmetry 2019, 11(6), 784; https://doi.org/10.3390/sym11060784 - 12 Jun 2019

Cited by 9 | Viewed by 2951

Abstract

In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of

κ

and

τ - 1

is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. [...] Read more.

In this article, we define Lorentzian cross product in a three-dimensional almost contact Lorentzian manifold. Using a Lorentzian cross product, we prove that the ratio of

κ

and

τ - 1

is constant along a Frenet slant curve in a Sasakian Lorentzian three-manifold. Moreover, we prove that

γ

is a slant curve if and only if M is Sasakian for a contact magnetic curve

γ

in contact Lorentzian three-manifold M. As an example, we find contact magnetic curves in Lorentzian Heisenberg three-space. Full article

(This article belongs to the Special Issue Geometry of Submanifolds and Homogeneous Spaces)

Search Results (4)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (4)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI