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Special (Pseudo-) Riemannian Manifolds

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: 30 April 2024 | Viewed by 6566

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Prof. Dr. Josef Mikeš

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Department of Algebra and Geometry, Palacky University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
Interests: differential geometry of (pseudo-) Riemannian manifolds and manifolds with connections; theory of geodesic, conformal, holomorphically-projective mappings of special manifolds geometry of (pseudo-) Riemannian manifolds and manifolds with connections; theory of geodesic, conformal, holomorphically-projective mappings of special manifolds
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Special Issue Information

Dear Colleagues,

Differential geometry studies several problems with applications in manifolds with Riemannian and other structures. The special manifolds play an important role in theoretical physics. Many issues arise in a local and global theory of special automorphisms, diffeomorphisms, and deformations that can be infinitesimal. The main theme of this Special Issue is differential geometric structures on manifolds and smooth maps that preserve these structures (e.g., geodesic, conformal, harmonic, holomorphically projective, rotary mappings, transformations, and deformations; special geometric vector fields; variational theory).

This Special Issue deals with the theory and applications of differential geometry and will accept original research papers. The purpose of this issue is to bring mathematicians together with physicists, as well as other scientists who use differential geometry as their research tool.

Prof. Dr. Josef Mikeš
Guest Editor

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Keywords

differentiable manifolds
(pseudo-)Riemannian geometry
geometry of spaces with structures
geodesics and their generalizations
differential invariants
variational theory
vector field
applications to physics
special mappings, transformations, and deformations
surfaces and special curves

Published Papers (7 papers)

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Research

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0 pages, 288 KiB

Open AccessArticle

Ricci Vector Fields Revisited

by Hanan Alohali, Sharief Deshmukh and Gabriel-Eduard Vîlcu

Mathematics 2024, 12(1), 144; https://0-doi-org.brum.beds.ac.uk/10.3390/math12010144 - 01 Jan 2024

Cited by 1 | Viewed by 682 | Correction

Abstract

We continue studying the

σ

-Ricci vector field

u

on a Riemannian manifold

(N^{m}, g)

, which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. [...] Read more.

We continue studying the

σ

-Ricci vector field

u

on a Riemannian manifold

(N^{m}, g)

, which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold

(N^{m}, g)

m > 1

, of positive scalar curvature

τ

, admits a closed

σ

-Ricci vector field

u

such that the vector

u - \nabla σ

is an eigenvector of T with eigenvalue

τ m^{- 1}

, if and only if it is isometric to the m-sphere

S_{α}^{m}

. In the second result, we show that if a compact and connected T-manifold

(N^{m}, g)

m > 2

, admits a

σ

-Ricci vector field

u

with

σ \neq 0

and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature

R i c (u, u)

that has a suitable lower bound, then

(N^{m}, g)

is isometric to the m-sphere

S_{α}^{m}

, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold

(N^{m}, g)

admits a

σ

-Ricci vector field

u

with

σ

as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature

R i c (u, u)

has a lower bound depending on a positive constant

α

, if and only if

(N^{m}, g)

is isometric to the m-sphere

S_{α}^{m}

. Full article

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

13 pages, 300 KiB

Open AccessArticle

Tensor Decompositions and Their Properties

by Patrik Peška, Marek Jukl and Josef Mikeš

Mathematics 2023, 11(17), 3638; https://0-doi-org.brum.beds.ac.uk/10.3390/math11173638 - 23 Aug 2023

Viewed by 670

Abstract

In the present paper, we study two different approaches of tensor decomposition. The first part aims to study some properties of tensors that result from the fact that some components are vanishing in certain coordinates. It is proven that these conditions allow tensor [...] Read more.

σ

σ = 1, 2, 3

tensors. We apply the results for special tensors such as the Riemann, Ricci, Einstein, and Weyl tensors and the deformation tensors of affine connections. Thereby, we find new criteria for the Einstein spaces, spaces of constant curvature, and projective and conformal flat spaces. Further, the proof of the theorem of Mikeš and Moldobayev is repaired. It has been used in many works and it is a generalization of the criteria formulated by Schouten and Struik. The second part deals with the properties of a special differential operator with respect to the general decomposition of tensor fields on manifolds with affine connection. It is shown that the properties of special differential operators are transferred to the components of a given decomposition. Full article

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

9 pages, 296 KiB

Open AccessArticle

Canonical F-Planar Mappings of Spaces with Affine Connection onto m-Symmetric Spaces

by Volodymyr Berezovski, Lenka Rýparová and Yevhen Cherevko

Mathematics 2023, 11(5), 1246; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051246 - 04 Mar 2023

Cited by 1 | Viewed by 784

Abstract

In this paper, we consider canonical F-planar mappings of spaces with affine connection onto m-symmetric spaces. We obtained the fundamental equations of these mappings in the form of a closed system of Chauchy-type equations in covariant derivatives. Furthermore, we established the [...] Read more.

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

13 pages, 310 KiB

Open AccessArticle

Infinitesimal Transformations of Riemannian Manifolds—The Geometric Dynamics Point of View

by Lenka Rýparová, Irena Hinterleitner, Sergey Stepanov and Irina Tsyganok

Mathematics 2023, 11(5), 1114; https://0-doi-org.brum.beds.ac.uk/10.3390/math11051114 - 23 Feb 2023

Cited by 1 | Viewed by 995

Abstract

In the present paper, we study the geometry of infinitesimal conformal, affine, projective, and harmonic transformations of complete Riemannian manifolds using the concepts of geometric dynamics and the methods of the modern version of the Bochner technique. Full article

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

14 pages, 315 KiB

Open AccessArticle

Poisson Doubly Warped Product Manifolds

by Ibrahim Al-Dayel, Foued Aloui and Sharief Deshmukh

Mathematics 2023, 11(3), 519; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030519 - 18 Jan 2023

Cited by 1 | Viewed by 1095

Abstract

This article generalizes some geometric structures on warped product manifolds equipped with a Poisson structure to doubly warped products of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. First, we introduce the notion of Poisson doubly warped product manifold [...] Read more.

(_{f} B \times_{b} F, Π = μ^{v} Π_{B}^{h} + ν^{h} Π_{F}^{v}, g)

and express the Levi-Civita contravariant connection, curvature and metacurvature of

(_{f} B \times_{b} F, Π, g)

in terms of Levi-Civita connections, curvatures and metacurvatures of components

(B, Π_{B}, g_{B})

and

(F, Π_{F}, g_{F})

. We also study compatibility conditions related to the Poisson structure

Π

and the contravariant metric g on

_{f} B \times_{b} F

, so that the compatibility conditions on

(B, Π_{B}, g_{B})

and

(F, Π_{F}, g_{F})

remain consistent in the Poisson doubly warped product manifold

(_{f} B \times_{b} F, Π, g)

. Full article

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

14 pages, 326 KiB

Open AccessArticle

A Contribution of Liouville-Type Theorems to the Geometry in the Large of Hadamard Manifolds

by Josef Mikeš, Vladimir Rovenski and Sergey Stepanov

Mathematics 2022, 10(16), 2880; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162880 - 11 Aug 2022

Viewed by 1296

Abstract

A complete, simply connected Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. In this article, we prove Liouville-type theorems for isometric and harmonic self-diffeomorphisms of Hadamard manifolds, as well as Liouville-type theorems for Killing–Yano, symmetric Killing and harmonic tensors on [...] Read more.

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

Other

Jump to: Research

1 pages, 133 KiB

Open AccessCorrection

Correction: Alohali et al. Ricci Vector Fields Revisited. Mathematics 2024, 12, 144

by Hanan Alohali, Sharief Deshmukh and Gabriel-Eduard Vîlcu

Mathematics 2024, 12(4), 512; https://0-doi-org.brum.beds.ac.uk/10.3390/math12040512 - 07 Feb 2024

Viewed by 323

Abstract

In the original paper [...] Full article

(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

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