Research

11 pages, 738 KiB

Open AccessArticle

Generalized Bertrand Curves in Minkowski 3-Space

by Chunxiao Zhang and Donghe Pei

Mathematics 2020, 8(12), 2199; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122199 - 10 Dec 2020

Cited by 5 | Viewed by 2432

We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the [...] Read more.

We define a generalized lightlike Bertrand curve pair and a generalized non-lightlike Bertrand curve pair, discuss their properties and prove the necessary and sufficient condition of a curve which is a generalized lightlike or a generalized non-lightlike Bertrand curve. Moreover, we study the relationship between slant helices and generalized Bertrand curves. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

7 pages, 274 KiB

Open AccessArticle

A Remark on Quadrics in Projective Klingenberg Spaces over a Certain Local Algebra

by Marek Jukl

Mathematics 2020, 8(12), 2168; https://0-doi-org.brum.beds.ac.uk/10.3390/math8122168 - 04 Dec 2020

Viewed by 1015

Abstract

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion “degree of neighborhood” is used [...] Read more.

This article is devoted to some polar properties of quadrics in the projective Klingenberg spaces over a local ring which is a linear algebra generated by one nilpotent element. In this case, polar subspaces are described; the notion “degree of neighborhood” is used for the geometric description of polar subspaces of quadrics. The polarity induced by a quadric is also studied. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

12 pages, 857 KiB

Open AccessArticle

Surfaces of Revolution and Canal Surfaces with Generalized Cheng–Yau 1-Type Gauss Maps

by Jinhua Qian, Xueshan Fu, Xueqian Tian and Young Ho Kim

Mathematics 2020, 8(10), 1728; https://0-doi-org.brum.beds.ac.uk/10.3390/math8101728 - 09 Oct 2020

Cited by 1 | Viewed by 1654

Abstract

In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. [...] Read more.

In the present work, the notion of generalized Cheng–Yau 1-type Gauss map is proposed, which is similar to the idea of generalized 1-type Gauss maps. Based on this concept, the surfaces of revolution and the canal surfaces in the Euclidean three-space are classified. First of all, we show that the Gauss map of any surfaces of revolution with a unit speed profile curve is of generalized Cheng–Yau 1-type. At the same time, an oriented canal surface has a generalized Cheng–Yau 1-type Gauss map if, and only if, it is an open part of a surface of revolution or a torus. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

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13 pages, 260 KiB

Open AccessArticle

Geodesic Mappings of Spaces with Affine Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

by Volodymyr Berezovski, Yevhen Cherevko, Irena Hinterleitner and Patrik Peška

Mathematics 2020, 8(9), 1560; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091560 - 11 Sep 2020

Cited by 8 | Viewed by 1600

Abstract

In the paper, we consider geodesic mappings of spaces with an affine connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an affine connections onto 2-, 3-, and m- (Ricci-) symmetric spaces. These [...] Read more.

In the paper, we consider geodesic mappings of spaces with an affine connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an affine connections onto 2-, 3-, and m- (Ricci-) symmetric spaces. These spaces play an important role in the General Theory of Relativity. The main results we obtained were generalized to a case of geodesic mappings of spaces with an affine connection onto (Ricci-) symmetric spaces. The main equations of the mappings were obtained as closed mixed systems of PDEs of the Cauchy type in covariant form. For the systems, we have found the maximum number of essential parameters which the solutions depend on. Any m- (Ricci-) symmetric spaces (

m \geq 1

) are geodesically mapped onto many spaces with an affine connection. We can call these spaces projectivelly m- (Ricci-) symmetric spaces and for them there exist above-mentioned nontrivial solutions. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

11 pages, 280 KiB

Open AccessArticle

Geodesic Vector Fields on a Riemannian Manifold

by Sharief Deshmukh, Patrik Peska and Nasser Bin Turki

Mathematics 2020, 8(1), 137; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010137 - 19 Jan 2020

Cited by 13 | Viewed by 5619

Abstract

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of [...] Read more.

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

9 pages, 265 KiB

Open AccessArticle

The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms

by Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok

Mathematics 2020, 8(1), 83; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010083 - 03 Jan 2020

Cited by 2 | Viewed by 1871

Abstract

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that [...] Read more.

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

15 pages, 1025 KiB

Open AccessArticle

Structure Functions of Pseudo Null Curves in Minkowski 3-Space

by Jinhua Qian, Jie Liu, Xueqian Tian and Young Ho Kim

Mathematics 2020, 8(1), 75; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010075 - 03 Jan 2020

Cited by 2 | Viewed by 2468

Abstract

In this work, the embankment surfaces with pseudo null base curves are investigated in Minkowski 3-space. The representation formula of pseudo null curves is obtained via the defined structure functions and the k-type pseudo null helices are discussed completely. Based on the theories [...] Read more.

In this work, the embankment surfaces with pseudo null base curves are investigated in Minkowski 3-space. The representation formula of pseudo null curves is obtained via the defined structure functions and the k-type pseudo null helices are discussed completely. Based on the theories of pseudo null curves, a class of embankment surfaces are constructed and characterized by the structure functions of the pseudo null base curves. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

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13 pages, 232 KiB

Open AccessArticle

Conformal Equitorsion and Concircular Transformations in a Generalized Riemannian Space

by Ana M. Velimirović

Mathematics 2020, 8(1), 61; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010061 - 02 Jan 2020

Cited by 5 | Viewed by 1595

Abstract

In the beginning, the basic facts about a conformal transformations are exposed and then equitorsion conformal transformations are defined. For every five independent curvature tensors in Generalized Riemannian space, the above cited transformations are investigated and corresponding invariants-5 concircular tensors of concircular transformations [...] Read more.

In the beginning, the basic facts about a conformal transformations are exposed and then equitorsion conformal transformations are defined. For every five independent curvature tensors in Generalized Riemannian space, the above cited transformations are investigated and corresponding invariants-5 concircular tensors of concircular transformations are found. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

8 pages, 280 KiB

Open AccessArticle

On Canonical Almost Geodesic Mappings of Type π₂(e)

by Volodymyr Berezovski, Josef Mikeš, Lenka Rýparová and Almazbek Sabykanov

Mathematics 2020, 8(1), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010054 - 01 Jan 2020

Cited by 8 | Viewed by 1451

Abstract

In the paper, we consider canonical almost geodesic mappings of type

π_{2} (e)

. We have found the conditions that must be satisfied for the mappings to preserve the Riemann tensor. Furthermore, we consider canonical almost geodesic mappings of type [...] Read more.

In the paper, we consider canonical almost geodesic mappings of type

π_{2} (e)

. We have found the conditions that must be satisfied for the mappings to preserve the Riemann tensor. Furthermore, we consider canonical almost geodesic mappings of type

π_{2} (e)

of spaces with affine connections onto symmetric spaces. The main equations for the mappings are obtained as a closed mixed system of Cauchy-type Partial Differential Equations. We have found the maximum number of essential parameters which the solution of the system depends on. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

15 pages, 298 KiB

Open AccessFeature PaperArticle

Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization

by Olga Belova

Mathematics 2019, 7(10), 901; https://0-doi-org.brum.beds.ac.uk/10.3390/math7100901 - 26 Sep 2019

Cited by 2 | Viewed by 1605

Abstract

The space

Π

of centered m-planes is considered in projective space

P_{n}

. A principal bundle is associated with the space

Π

and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization [...] Read more.

The space

Π

of centered m-planes is considered in projective space

P_{n}

. A principal bundle is associated with the space

Π

and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space

Π

. Semi-normalized spaces

Π^{1}

,

Π^{2}

and normalized space

Π^{1, 2}

are investigated. By virtue of the Cartan–Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space

Π

to the normalized space

Π^{1, 2}

. Full article

(This article belongs to the Special Issue Differential Geometry of Spaces with Structures)

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