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Mathematics
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Special (Pseudo-) Riemannian Manifolds
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Special (Pseudo-) Riemannian Manifolds

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

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 8187

Special Issue Editor

Prof. Dr. Josef Mikeš

E-Mail Website
Guest Editor
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 and other special mappings, transformations and deformations of manifolds
Special Issues, Collections and Topics in MDPI journals

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

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Published Papers (6 papers)

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Research

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16 pages, 288 KiB  
Article
Ricci Vector Fields Revisited
by Hanan Alohali, Sharief Deshmukh and Gabriel-Eduard Vîlcu
Mathematics 2024, 12(1), 144; https://doi.org/10.3390/math12010144 - 1 Jan 2024
Cited by 1 | Viewed by 1074 | Correction
Abstract
We continue studying the σ-Ricci vector field u on a Riemannian manifold (Nm,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 (Nm,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(Nm,g), m>1, of positive scalar curvature τ, admits a closed σ-Ricci vector field u such that the vector u−∇σ is an eigenvector of T with eigenvalue τm−1, if and only if it is isometric to the m-sphere Sαm. In the second result, we show that if a compact and connected T-manifold(Nm,g), m>2, admits a σ-Ricci vector field u with σ≠0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature Ricu,u that has a suitable lower bound, then (Nm,g) is isometric to the m-sphere Sαm, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold (Nm,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 Ricu,u has a lower bound depending on a positive constant α, if and only if (Nm,g) is isometric to the m-sphere Sαm. Full article
(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)
13 pages, 300 KiB  
Article
Tensor Decompositions and Their Properties
by Patrik Peška, Marek Jukl and Josef Mikeš
Mathematics 2023, 11(17), 3638; https://doi.org/10.3390/math11173638 - 23 Aug 2023
Cited by 2 | Viewed by 1114
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.
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 decomposition, especially (1, 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  
Article
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://doi.org/10.3390/math11051246 - 4 Mar 2023
Cited by 1 | Viewed by 1200
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.
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 number of essential parameters on which its general solution depends. Full article
(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)
13 pages, 310 KiB  
Article
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://doi.org/10.3390/math11051114 - 23 Feb 2023
Cited by 1 | Viewed by 1630
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  
Article
Poisson Doubly Warped Product Manifolds
by Ibrahim Al-Dayel, Foued Aloui and Sharief Deshmukh
Mathematics 2023, 11(3), 519; https://doi.org/10.3390/math11030519 - 18 Jan 2023
Cited by 3 | Viewed by 1670
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.
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 (fB×bF,Π=μvΠBh+νhΠFv,g) and express the Levi-Civita contravariant connection, curvature and metacurvature of (fB×bF,Π,g) in terms of Levi-Civita connections, curvatures and metacurvatures of components (B,ΠB,gB) and (F,ΠF,gF). We also study compatibility conditions related to the Poisson structure Π and the contravariant metric g on fB×bF, so that the compatibility conditions on (B,ΠB,gB) and (F,ΠF,gF) remain consistent in the Poisson doubly warped product manifold (fB×bF,Π,g). Full article
(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)

Other

Jump to: Research

1 pages, 133 KiB  
Correction
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://doi.org/10.3390/math12040512 - 7 Feb 2024
Viewed by 591
Abstract
In the original paper [...] Full article
(This article belongs to the Special Issue Special (Pseudo-) Riemannian Manifolds)
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