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17 pages, 512 KiB

Open AccessArticle

R

⁷

by Ana Irina Nistor

Mathematics 2022, 10(19), 3480; https://doi.org/10.3390/math10193480 - 23 Sep 2022

Viewed by 1114

Abstract

In this paper we study the magnetic trajectories as the solutions of the Lorentz equation defined by the cross product corresponding to the 7-dimensional Euclidean space. We find several examples of such trajectories and moreover, we strongly motivate our results making a comparison [...] Read more.

In this paper we study the magnetic trajectories as the solutions of the Lorentz equation defined by the cross product corresponding to the 7-dimensional Euclidean space. We find several examples of such trajectories and moreover, we strongly motivate our results making a comparison with the 3-dimensional Euclidean case, ambient space which was among the first ones approached in the study of magnetic trajectories. Full article

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

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18 pages, 339 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Almost Complex and Hypercomplex Norden Structures Induced by Natural Riemann Extensions

by Cornelia-Livia Bejan and Galia Nakova

Mathematics 2022, 10(15), 2625; https://doi.org/10.3390/math10152625 - 27 Jul 2022

Cited by 1 | Viewed by 1463

Abstract

The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle

T^{*} M

of a smooth manifold M, induced by a symmetric linear connection ∇ on M. In [...] Read more.

The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle

T^{*} M

of a smooth manifold M, induced by a symmetric linear connection ∇ on M. In this paper we deal with a natural Riemann extension

\bar{g}

, which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure

\bar{J}

on the cotangent bundle

T^{*} M

of an almost complex manifold

(M, J, \nabla)

with a symmetric linear connection ∇ such that

(T^{*} M, \bar{J}, \bar{g})

is an almost complex manifold, where the natural Riemann extension

\bar{g}

is a Norden metric. We obtain necessary and sufficient conditions for

(T^{*} M, \bar{J}, \bar{g})

to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold

(M, J, \nabla^{'})

endowed with an almost complex connection

\nabla^{'}

(

\nabla^{'} J = 0

). We investigate the harmonicity with respect to

\bar{g}

of the almost complex structure

\bar{J}

, according to the type of the base manifold. Moreover, we define an almost hypercomplex structure

({\bar{J}}_{1}, {\bar{J}}_{2}, {\bar{J}}_{3})

on the cotangent bundle

T^{*} M^{4 n}

of an almost hypercomplex manifold

(M^{4 n}, J_{1}, J_{2}, J_{3}, \nabla)

with a symmetric linear connection ∇. The natural Riemann extension

\bar{g}

is a Hermitian metric with respect to

{\bar{J}}_{1}

and a Norden metric with respect to

{\bar{J}}_{2}

and

{\bar{J}}_{3}

. Full article

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

19 pages, 1030 KiB

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 1 | Viewed by 3165

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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21 pages, 656 KiB

Open AccessArticle

Magnetic Geodesic in (Almost) Cosymplectic Lie Groups of Dimension 3

by Marian Ioan Munteanu

Mathematics 2022, 10(4), 544; https://doi.org/10.3390/math10040544 - 10 Feb 2022

Viewed by 1371

Abstract

In this paper, we study contact magnetic geodesics in a 3-dimensional Lie group G endowed with a left invariant almost cosymplectic structure. We distinguish the two cases: G is unimodular, and G is nonunimodular. We pay a careful attention to the special case [...] Read more.

In this paper, we study contact magnetic geodesics in a 3-dimensional Lie group G endowed with a left invariant almost cosymplectic structure. We distinguish the two cases: G is unimodular, and G is nonunimodular. We pay a careful attention to the special case where the structure is cosymplectic, and we write down explicit expressions of magnetic geodesics and corresponding magnetic Jacobi fields. Full article

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

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40 pages, 463 KiB

Open AccessArticle

Beltrami Equations on Rossi Spheres

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Mathematics 2022, 10(3), 371; https://doi.org/10.3390/math10030371 - 25 Jan 2022

Viewed by 1964

Abstract

Beltrami equations

{\bar{L}}_{t} (g) = μ (\cdot, t) L_{t} (g)

on

S^{3}

(where

L_{t}

,

| t | < 1

, are the Rossi operators i.e.,

L_{t}

spans the globally [...] Read more.

Beltrami equations

{\bar{L}}_{t} (g) = μ (\cdot, t) L_{t} (g)

on

S^{3}

(where

L_{t}

,

| t | < 1

, are the Rossi operators i.e.,

L_{t}

spans the globally nonembeddable CR structure

H (t)

on

S^{3}

discovered by H. Rossi) are derived such that to describe quasiconformal mappings

f : S^{3} \to N \subset C^{2}

from the Rossi sphere

(S^{3}, H (t))

. Using the Greiner–Kohn–Stein solution to the Lewy equation and the Bargmann representations of the Heisenberg group, we solve the Beltrami equations for Sobolev-type solutions

g_{t}

such that

g_{t} - v \in W_{F}^{1, 2} (S^{3}, θ)

with

v \in {CR}^{\infty} (S^{3}, H (0))

. Full article

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

11 pages, 853 KiB

Open AccessArticle

On the Geometrical Properties of the Lightlike Rectifying Curves and the Centrodes

by Jianguo Sun, Yanping Zhao and Xiaoyan Jiang

Mathematics 2021, 9(23), 3103; https://doi.org/10.3390/math9233103 - 1 Dec 2021

Cited by 2 | Viewed by 1418

Abstract

This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves [...] Read more.

This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves are the lightlike rectifying curves. Meanwhile, a detailed analysis between the N-type lightlike slant helices and the centrodes of lightlike curves is provided in this paper. We give the projections of the lightlike rectifying curves to the timelike planes. Full article

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

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11 pages, 322 KiB

Open AccessArticle

Characterization of Rectifying Curves by Their Involutes and Evolutes

by Marilena Jianu, Sever Achimescu, Leonard Dăuş, Adela Mihai, Olimpia-Alice Roman and Daniel Tudor

Mathematics 2021, 9(23), 3077; https://doi.org/10.3390/math9233077 - 29 Nov 2021

Cited by 7 | Viewed by 3042

Abstract

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in [...] Read more.

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves. Full article

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

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11 pages, 824 KiB

Open AccessArticle

Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

by Xin Zhao and Donghe Pei

Mathematics 2021, 9(22), 2852; https://doi.org/10.3390/math9222852 - 10 Nov 2021

Cited by 8 | Viewed by 1856

Abstract

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane

R_{1}^{2}

. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For [...] Read more.

In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane

R_{1}^{2}

. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve. Full article

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

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9 pages, 250 KiB

Open AccessArticle

A Characterization of GRW Spacetimes

by Ibrahim Al-Dayel, Sharief Deshmukh and Mohd. Danish Siddiqi

Mathematics 2021, 9(18), 2209; https://doi.org/10.3390/math9182209 - 9 Sep 2021

Cited by 6 | Viewed by 1628

Abstract

We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like [...] Read more.

We show presence a special torse-forming vector field (a particular form of torse-forming of a vector field) on generalized Robertson–Walker (GRW) spacetime, which is an eigenvector of the de Rham–Laplace operator. This paves the way to showing that the presence of a time-like special torse-forming vector field

ξ

with potential function

ρ

on a Lorentzian manifold

(M, g)

, dim

M > 5

, which is an eigenvector of the de Rham Laplace operator, gives a characterization of a GRW-spacetime. We show that if, in addition, the function

ξ (ρ)

is nowhere zero, then the fibers of the GRW-spacetime are compact. Finally, we show that on a simply connected Lorentzian manifold

(M, g)

that admits a time-like special torse-forming vector field

ξ

, there is a function f called the associated function of

ξ

. It is shown that if a connected Lorentzian manifold

(M, g)

, dim

M > 4

, admits a time-like special torse-forming vector field

ξ

with associated function f nowhere zero and satisfies the Fischer–Marsden equation, then

(M, g)

is a quasi-Einstein manifold. Full article

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

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