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3 pages, 158 KiB

Open AccessEditorial

Preface to: Differential Geometry: Structures on Manifolds and Their Applications

by Marian Ioan Munteanu

Mathematics 2022, 10(13), 2243; https://doi.org/10.3390/math10132243 - 27 Jun 2022

Viewed by 1466

Abstract

When a manifold is endowed with a geometric structure, we have more opportunities to explore its geometric properties [...] Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

Research

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21 pages, 336 KiB

Open AccessArticle

Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere

by Masatomo Takahashi and Haiou Yu

Mathematics 2022, 10(8), 1292; https://doi.org/10.3390/math10081292 - 13 Apr 2022

Cited by 3 | Viewed by 1524

Abstract

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical [...] Read more.

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

18 pages, 359 KiB

Open AccessFeature PaperArticle

Magnetic Jacobi Fields in 3-Dimensional Cosymplectic Manifolds

by Marian Ioan Munteanu and Ana Irina Nistor

Mathematics 2021, 9(24), 3220; https://doi.org/10.3390/math9243220 - 13 Dec 2021

Cited by 5 | Viewed by 1814

Abstract

We classify the magnetic Jacobi fields in cosymplectic manifolds of dimension 3, enriching the results in the study of magnetic Jacobi fields derived from uniform magnetic fields. In particular, we give examples of Jacobi magnetic fields in the Euclidean space

E^{3}

and [...] Read more.

We classify the magnetic Jacobi fields in cosymplectic manifolds of dimension 3, enriching the results in the study of magnetic Jacobi fields derived from uniform magnetic fields. In particular, we give examples of Jacobi magnetic fields in the Euclidean space

E^{3}

and we conclude with the description of magnetic Jacobi fields in the product spaces

S^{2} \times R

and

H^{2} \times R

. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

12 pages, 273 KiB

Open AccessArticle

On an Anti-Torqued Vector Field on Riemannian Manifolds

by Sharief Deshmukh, Ibrahim Al-Dayel and Devaraja Mallesha Naik

Mathematics 2021, 9(18), 2201; https://doi.org/10.3390/math9182201 - 8 Sep 2021

Cited by 4 | Viewed by 1501

Abstract

A torqued vector field

ξ

is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is [...] Read more.

A torqued vector field

ξ

is a torse-forming vector field on a Riemannian manifold that is orthogonal to the dual vector field of the 1-form in the definition of torse-forming vector field. In this paper, we introduce an anti-torqued vector field which is opposite to torqued vector field in the sense it is parallel to the dual vector field to the 1-form in the definition of torse-forming vector fields. It is interesting to note that anti-torqued vector fields do not reduce to concircular vector fields nor to Killing vector fields and thus, give a unique class among the classes of special vector fields on Riemannian manifolds. These vector fields do not exist on compact and simply connected Riemannian manifolds. We use anti-torqued vector fields to find two characterizations of Euclidean spaces. Furthermore, a characterization of an Einstein manifold is obtained using the combination of a torqued vector field and Fischer–Marsden equation. We also find a condition under which the scalar curvature of a compact Riemannian manifold admitting an anti-torqued vector field is strictly negative. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

15 pages, 293 KiB

Open AccessArticle

Warped Product Submanifolds in Locally Golden Riemannian Manifolds with a Slant Factor

by Cristina E. Hretcanu and Adara M. Blaga

Mathematics 2021, 9(17), 2125; https://doi.org/10.3390/math9172125 - 2 Sep 2021

Cited by 3 | Viewed by 1841

Abstract

In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in [...] Read more.

In the present paper, we study some properties of warped product pointwise semi-slant and hemi-slant submanifolds in Golden Riemannian manifolds, and we construct examples in Euclidean spaces. Additionally, we study some properties of proper warped product pointwise semi-slant (and, respectively, hemi-slant) submanifolds in a locally Golden Riemannian manifold. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

26 pages, 2621 KiB

Open AccessArticle

Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

by Federico Zadra, Alessandro Bravetti and Marcello Seri

Mathematics 2021, 9(16), 1960; https://doi.org/10.3390/math9161960 - 16 Aug 2021

Cited by 5 | Viewed by 2874

Abstract

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and [...] Read more.

Starting from a contact Hamiltonian description of Liénard systems, we introduce a new family of explicit geometric integrators for these nonlinear dynamical systems. Focusing on the paradigmatic example of the van der Pol oscillator, we demonstrate that these integrators are particularly stable and preserve the qualitative features of the dynamics, even for relatively large values of the time step and in the stiff regime. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

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10 pages, 260 KiB

Open AccessArticle

Some Conditions on Trans-Sasakian Manifolds to Be Homothetic to Sasakian Manifolds

by Sharief Deshmukh, Amira Ishan, Olga Belova and Suha B. Al-Shaikh

Mathematics 2021, 9(16), 1887; https://doi.org/10.3390/math9161887 - 8 Aug 2021

Cited by 5 | Viewed by 1818

Abstract

In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and [...] Read more.

In this paper, we study 3-dimensional compact and connected trans-Sasakian manifolds and find necessary and sufficient conditions under which these manifolds are homothetic to Sasakian manifolds. First, four results in this paper deal with finding necessary and sufficient conditions on a compact and connected trans-Sasakian manifold to be homothetic to a compact and connected Sasakian manifold, and the fifth result deals with finding necessary and sufficient condition on a connected trans-Sasakian manifold to be homothetic to a connected Sasakian manifold. Finally, we find necessary and sufficient conditions on a compact and simply connected trans-Sasakian manifold to be homothetic to a compact and simply connected Einstein Sasakian manifold. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

17 pages, 489 KiB

Open AccessArticle

Principal Bundle Structure of Matrix Manifolds

by Marie Billaud-Friess, Antonio Falcó and Anthony Nouy

Mathematics 2021, 9(14), 1669; https://doi.org/10.3390/math9141669 - 15 Jul 2021

Cited by 2 | Viewed by 2059

Abstract

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold

G_{r} (R^{k})

of linear subspaces of dimension

r < k

in [...] Read more.

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold

G_{r} (R^{k})

of linear subspaces of dimension

r < k

in

R^{k}

, which avoids the use of equivalence classes. The set

G_{r} (R^{k})

is equipped with an atlas, which provides it with the structure of an analytic manifold modeled on

R^{(k - r) \times r}

. Then, we define an atlas for the set

M_{r} (R^{k \times r})

of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base

G_{r} (R^{k})

and typical fibre

{GL}_{r}

, the general linear group of invertible matrices in

R^{k \times k}

. Finally, we define an atlas for the set

M_{r} (R^{n \times m})

of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base

G_{r} (R^{n}) \times G_{r} (R^{m})

and typical fibre

{GL}_{r}

. The atlas of

M_{r} (R^{n \times m})

is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set

M_{r} (R^{n \times m})

equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space

R^{n \times m}

equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space

R^{n \times m}

, seen as the union of manifolds

M_{r} (R^{n \times m})

, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

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12 pages, 265 KiB

Open AccessArticle

Representations of Rectifying Isotropic Curves and Their Centrodes in Complex 3-Space

by Jinhua Qian, Pei Yin, Xueshan Fu and Hongzeng Wang

Mathematics 2021, 9(12), 1451; https://doi.org/10.3390/math9121451 - 21 Jun 2021

Cited by 2 | Viewed by 1639

Abstract

In this work, the rectifying isotropic curves are investigated in three-dimensional complex space

C^{3}

. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. [...] Read more.

In this work, the rectifying isotropic curves are investigated in three-dimensional complex space

C^{3}

. The conclusion that an isotropic curve is a rectifying curve if and only if its pseudo curvature is a linear function of its pseudo arc-length is achieved. Meanwhile, the rectifying isotropic curves are expressed by the Bessel functions explicitly. Last but not least, the centrodes of rectifying isotropic curves are explored in detail. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

9 pages, 826 KiB

Open AccessArticle

Geometrical Properties of the Pseudonull Hypersurfaces in Semi-Euclidean 4-Space

by Jianguo Sun, Xiaoyan Jiang and Fenghui Ji

Mathematics 2021, 9(11), 1274; https://doi.org/10.3390/math9111274 - 1 Jun 2021

Cited by 5 | Viewed by 2298

Abstract

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example [...] Read more.

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

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14 pages, 548 KiB

Open AccessArticle

Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

by Rafael López, Željka Milin Šipuš, Ljiljana Primorac Gajčić and Ivana Protrka

Mathematics 2021, 9(11), 1256; https://doi.org/10.3390/math9111256 - 31 May 2021

Cited by 9 | Viewed by 3156

Abstract

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. [...] Read more.

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

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12 pages, 297 KiB

Open AccessEditor’s ChoiceArticle

Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces

by Volodymyr Berezovski, Yevhen Cherevko, Josef Mikeš and Lenka Rýparová

Mathematics 2021, 9(4), 437; https://doi.org/10.3390/math9040437 - 22 Feb 2021

Cited by 10 | Viewed by 2099

Abstract

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed [...] Read more.

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš. Full article

(This article belongs to the Special Issue Differential Geometry: Structures on Manifolds and Their Applications)

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