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20 pages, 291 KB

Open AccessArticle

Half-Symmetric Connections of Generalized Riemannian Spaces

by Marko Stefanović, Mića S. Stanković, Ivana Djurišić and Nenad Vesić

Axioms 2025, 14(12), 923; https://doi.org/10.3390/axioms14120923 - 16 Dec 2025

Viewed by 261

In this article, we generalize Yano’s concept of a half-symmetric affine connection. With respect to this generalization, we obtain five linearly independent curvature tensors. In the following, we examine which special kinds of affine connections may be the generalized half-symmetric affine connection. At the end of this work, we generalize the term of Killing’s vector given by Yano to affine Killing, conformal Killing, projective Killing, harmonic, and covariant and contravariant analytic vectors. Full article

(This article belongs to the Special Issue Advances in Geometry and Its Applications)

16 pages, 323 KB

Open AccessArticle

Metallic Structures for Tangent Bundles over Almost Quadratic ϕ-Manifolds

by Mohammad Nazrul Islam Khan, Sudhakar Kumar Chaubey, Nahid Fatima and Afifah Al Eid

Mathematics 2023, 11(22), 4683; https://doi.org/10.3390/math11224683 - 17 Nov 2023

Cited by 3 | Viewed by 1432

Abstract

This paper aims to explore the metallic structure

J^{2} = p J + q I,

where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle

T M

over almost quadratic

ϕ

-structures (briefly, [...] Read more.

This paper aims to explore the metallic structure

J^{2} = p J + q I,

where p and q are natural numbers, using complete and horizontal lifts on the tangent bundle

T M

over almost quadratic

ϕ

-structures (briefly,

(ϕ, ξ, η)

). Tensor fields

\tilde{F}

and

F^{*}

are defined on

T M

, and it is shown that they are metallic structures over

(ϕ, ξ, η)

. Next, the fundamental 2-form

Ω

and its derivative

d Ω

, with the help of complete lift on

T M

over

(ϕ, ξ, η)

, are evaluated. Furthermore, the integrability conditions and expressions of the Lie derivative of metallic structures

\tilde{F}

and

F^{*}

are determined using complete and horizontal lifts on

T M

over

(ϕ, ξ, η)

, respectively. Finally, we prove the existence of almost quadratic

ϕ

-structures on

T M

with non-trivial examples. Full article

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

12 pages, 288 KB

Open AccessArticle

Characterizations of the Frame Bundle Admitting Metallic Structures on Almost Quadratic ϕ-Manifolds

by Mohammad Nazrul Islam Khan, Uday Chand De and Teg Alam

Mathematics 2023, 11(14), 3097; https://doi.org/10.3390/math11143097 - 13 Jul 2023

Cited by 3 | Viewed by 1322

Abstract

In this work, we have characterized the frame bundle

F M

admitting metallic structures on almost quadratic

ϕ

-manifolds

ϕ^{2} = p ϕ + q I - q η \otimes ζ

, where p is an arbitrary constant and q is a [...] Read more.

In this work, we have characterized the frame bundle

F M

admitting metallic structures on almost quadratic

ϕ

-manifolds

ϕ^{2} = p ϕ + q I - q η \otimes ζ

, where p is an arbitrary constant and q is a nonzero constant. The complete lifts of an almost quadratic

ϕ

-structure to the metallic structure on

F M

are constructed. We also prove the existence of a metallic structure on

F M

with the aid of the

\tilde{J}

tensor field, which we define. Results for the 2-Form and its derivative are then obtained. Additionally, we derive the expressions of the Nijenhuis tensor of a tensor field

\tilde{J}

F M

. Finally, we construct an example of it to finish. Full article

(This article belongs to the Special Issue Geometry of Manifolds and Applications)

10 pages, 267 KB

Open AccessArticle

Kerr–Schild Tetrads and the Nijenhuis Tensor

by José Wadih Maluf, Fernando Lessa Carneiro, Sérgio Ulhoa and José Francisco Da Rocha-Neto

Universe 2023, 9(3), 127; https://doi.org/10.3390/universe9030127 - 28 Feb 2023

Viewed by 1751

Abstract

We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor

S_{μ}^{λ}

. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, [...] Read more.

We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor

S_{μ}^{λ}

. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, and (ii) the Minkowski space–time metric tensor into a non-flat metric tensor, and vice-versa. The

S_{μ}^{λ}

tensor and its inverse are constructed in terms of the standard null vector field

l_{μ}

that defines the Kerr–Schild form of the metric tensor in general relativity, and that yields black holes and non-linear gravitational waves as solutions of the vacuum Einstein’s field equations. We demonstrate that the condition for the vanishing of the Ricci tensor obtained by Kerr and Schild, in empty space–time, is also a condition for the vanishing of the Nijenhuis tensor constructed out of

S_{μ}^{λ}

. Thus, a theory based on the Nijenhuis tensor yields an important class of solutions of the Einstein’s field equations, namely, black holes and non-linear gravitational waves. We also demonstrate that the present mathematical framework can easily admit modifications of the Newtonian potential that may explain the long range gravitational effects related to galaxy rotation curves. Full article

(This article belongs to the Section Gravitation)

9 pages, 266 KB

Open AccessArticle

Nearly Sasakian Manifolds of Constant Type

by Aligadzhi Rustanov

Axioms 2022, 11(12), 673; https://doi.org/10.3390/axioms11120673 - 26 Nov 2022

Viewed by 1830

Abstract

The article deals with nearly Sasakian manifolds of a constant type. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the nearly Sasakian manifold is a nearly Kähler manifold. It is proved that the class of nearly Sasakian manifolds of the zero constant type coincides with the class of Sasakian manifolds. The concept of constancy of the type of an almost contact metric manifold is introduced through its Nijenhuis tensor, and the criterion of constancy of the type of an almost contact metric manifold is proved. The coincidence of both concepts of type constancy for the nearly Sasakian manifold is proved. It is proved that the almost Hermitian structure induced on the integral manifolds of the maximum dimension of the first fundamental distribution of the almost contact metric manifold of the zero constant type is the Hermitian structure. Full article

(This article belongs to the Special Issue Mathematical Analysis Is the Theoretical Basis of a Number of Mathematical Disciplines)

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