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Mathematics
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Differential Geometry of Special Mappings
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Differential Geometry of Special Mappings

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 July 2019) | Viewed by 26156

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 mappings of special manifolds geometry of (pseudo-) Riemannian manifolds and manifolds with connections; theory of geodesic, conformal, holomorphically-projective mappings of special manifolds
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is very well known that differential geometry studies a number of interesting problems, and the geometry has very applicable potential. There are many applications to (pseudo-) Riemannian and Finsler geometry, and also to the geometry of manifolds with affine and projective connections (e.g., special mappings of manifolds–geodesic, conformal, holomorphically-projective mappings, transformations and deformations), variational theory and physics.

The purpose of this Special Issue is to bring mathematicians together with physicists, as well as other scientists, for whom differential geometry is a valuable research tool.

This Special Issue deals with the theory and applications of differential geometry, especially in physics, and will accept high-quality papers having original research results. The Guest Editor solicits papers dealing with these challenging questions in the language of mathematics.

Prof. Dr. Josef Mikeš
Guest Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Differentiable manifolds
  • Geometry of spaces with structures
  • (pseudo-) Riemannian geometry
  • Geodesics and their generalizations
  • Special mappings and transformations
  • Differential invariants
  • Variational theory on manifolds
  • Applications to physics

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Published Papers (10 papers)

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Research

12 pages, 238 KiB  
Article
A Note on Minimal Translation Graphs in Euclidean Space
by Dan Yang, Jingjing Zhang and Yu Fu
Mathematics 2019, 7(10), 889; https://doi.org/10.3390/math7100889 - 24 Sep 2019
Cited by 7 | Viewed by 2503
Abstract
In this note, we give a characterization of a class of minimal translation graphs generated by planar curves. Precisely, we prove that a hypersurface that can be written as the sum of n planar curves is either a hyperplane or a cylinder on [...] Read more.
In this note, we give a characterization of a class of minimal translation graphs generated by planar curves. Precisely, we prove that a hypersurface that can be written as the sum of n planar curves is either a hyperplane or a cylinder on the generalized Scherk surface. This result can be considered as a generalization of the results on minimal translation hypersurfaces due to Dillen–Verstraelen–Zafindratafa in 1991 and minimal translation surfaces due to Liu–Yu in 2013. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
6 pages, 247 KiB  
Article
There Are No Conformal Einstein Rescalings of Pseudo-Riemannian Einstein Spaces with n Complete Light-Like Geodesics
by Josef Mikeš, Irena Hinterleitner and Nadezda Guseva
Mathematics 2019, 7(9), 801; https://doi.org/10.3390/math7090801 - 1 Sep 2019
Cited by 11 | Viewed by 2198
Abstract
In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which [...] Read more.
In the present paper, we study conformal mappings between a connected n-dimension pseudo-Riemannian Einstein manifolds. Let g be a pseudo-Riemannian Einstein metric of indefinite signature on a connected n-dimensional manifold M. Further assume that there is a point at which not all sectional curvatures are equal and through which in linearly independent directions pass n complete null (light-like) geodesics. If, for the function ψ the metric ψ − 2 g is also Einstein, then ψ is a constant, and conformal mapping is homothetic. Note that Kiosak and Matveev previously assumed that all light-lines were complete. If the Einstein manifold is closed, the completeness assumption can be omitted (the latter result is due to Mikeš and Kühnel). Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
8 pages, 300 KiB  
Article
A Remark on Structure of Projective Klingenberg Spaces over a Certain Local Algebra
by Marek Jukl
Mathematics 2019, 7(8), 702; https://doi.org/10.3390/math7080702 - 3 Aug 2019
Cited by 1 | Viewed by 2231
Abstract
This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using [...] Read more.
This article is devoted to the projective Klingenberg spaces over a local ring, which is a linear algebra generated by one nilpotent element. In this case, subspaces of such Klingenberg spaces are described. The notion of the “degree of neighborhood” is introduced. Using this, we present the geometric description of subsets of points of a projective Klingenberg space whose arithmetical representatives need not belong to a free submodule. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
11 pages, 307 KiB  
Article
Geodesic Mappings of Vn(K)-Spaces and Concircular Vector Fields
by Igor G. Shandra and Josef Mikeš
Mathematics 2019, 7(8), 692; https://doi.org/10.3390/math7080692 - 1 Aug 2019
Cited by 6 | Viewed by 2670
Abstract
In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called V n ( K ) -spaces. We prove that the set of solutions of the system of equations of geodesic mappings on V n ( K ) -spaces forms a [...] Read more.
In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called V n ( K ) -spaces. We prove that the set of solutions of the system of equations of geodesic mappings on V n ( K ) -spaces forms a special Jordan algebra and the set of solutions generated by concircular fields is an ideal of this algebra. We show that pseudo-Riemannian manifolds admitting a concircular field of the basic type form the class of manifolds closed with respect to the geodesic mappings. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
8 pages, 235 KiB  
Article
Conformal and Geodesic Mappings onto Some Special Spaces
by Volodymyr Berezovski, Yevhen Cherevko and Lenka Rýparová
Mathematics 2019, 7(8), 664; https://doi.org/10.3390/math7080664 - 25 Jul 2019
Cited by 6 | Viewed by 2997
Abstract
In this paper, we consider conformal mappings of Riemannian spaces onto Ricci-2-symmetric Riemannian spaces and geodesic mappings of spaces with affine connections onto Ricci-2-symmetric spaces. The main equations for the mappings are obtained as a closed system of Cauchy-type differential equations in covariant [...] Read more.
In this paper, we consider conformal mappings of Riemannian spaces onto Ricci-2-symmetric Riemannian spaces and geodesic mappings of spaces with affine connections onto Ricci-2-symmetric spaces. The main equations for the mappings are obtained as a closed system of Cauchy-type differential equations in covariant derivatives. We find the number of essential parameters which the solution of the system depends on. A similar approach was applied for the case of conformal mappings of Riemannian spaces onto Ricci-m-symmetric Riemannian spaces, as well as geodesic mappings of spaces with affine connections onto Ricci-m-symmetric spaces. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
16 pages, 304 KiB  
Article
Infinitesimal Transformations of Locally Conformal Kähler Manifolds
by Yevhen Cherevko, Volodymyr Berezovski, Irena Hinterleitner and Dana Smetanová
Mathematics 2019, 7(8), 658; https://doi.org/10.3390/math7080658 - 24 Jul 2019
Cited by 1 | Viewed by 2769
Abstract
The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also [...] Read more.
The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
10 pages, 237 KiB  
Article
On Conformal and Concircular Diffeomorphisms of Eisenhart’s Generalized Riemannian Spaces
by Miloš Z. Petrović, Mića S. Stanković and Patrik Peška
Mathematics 2019, 7(7), 626; https://doi.org/10.3390/math7070626 - 15 Jul 2019
Cited by 6 | Viewed by 2158
Abstract
We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces. We prove conformal and concircular invariance of some tensors in Eisenhart’s generalized Riemannian spaces. We give new generalizations of symmetric spaces via Eisenhart’s generalized Riemannian spaces. Finally, we describe some properties of [...] Read more.
We consider conformal and concircular mappings of Eisenhart’s generalized Riemannian spaces. We prove conformal and concircular invariance of some tensors in Eisenhart’s generalized Riemannian spaces. We give new generalizations of symmetric spaces via Eisenhart’s generalized Riemannian spaces. Finally, we describe some properties of covariant derivatives of tensors analogous to Yano’s tensor of concircular curvature in Eisenhart symmetric spaces of various kinds. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
8 pages, 274 KiB  
Article
The Mixed Scalar Curvature of a Twisted Product Riemannian Manifolds and Projective Submersions
by Vladimir Rovenski, Sergey Stepanov and Irina Tsyganok
Mathematics 2019, 7(6), 527; https://doi.org/10.3390/math7060527 - 10 Jun 2019
Viewed by 2278
Abstract
In the present paper, we study twisted and warped products of Riemannian manifolds. As an application, we consider projective submersions of Riemannian manifolds, since any Riemannian manifold admitting a projective submersion is necessarily a twisted product of some two Riemannian manifolds. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
11 pages, 283 KiB  
Article
Classification of Warped Product Submanifolds in Kenmotsu Space Forms Admitting Gradient Ricci Solitons
by Ali H. Alkhaldi and Akram Ali
Mathematics 2019, 7(2), 112; https://doi.org/10.3390/math7020112 - 22 Jan 2019
Cited by 10 | Viewed by 2471
Abstract
The purpose of this article is to obtain geometric conditions in terms of gradient Ricci curvature, both necessary and sufficient, for a warped product semi-slant in a Kenmotsu space form, to be either CR-warped product or simply a Riemannian product manifold when a [...] Read more.
The purpose of this article is to obtain geometric conditions in terms of gradient Ricci curvature, both necessary and sufficient, for a warped product semi-slant in a Kenmotsu space form, to be either CR-warped product or simply a Riemannian product manifold when a basic inequality become equality. The next purpose of this paper to find the necessary condition admitting gradient Ricci soliton, that the warped product semi-slant submanifold of Kenmotsu space form, is an Einstein warped product. We also discuss some obstructions to these constructions in more detail. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
15 pages, 247 KiB  
Article
Higher Order Hamiltonian Systems with Generalized Legendre Transformation
by Dana Smetanová
Mathematics 2018, 6(9), 163; https://doi.org/10.3390/math6090163 - 10 Sep 2018
Cited by 3 | Viewed by 3004
Abstract
The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton [...] Read more.
The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler–Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition. Full article
(This article belongs to the Special Issue Differential Geometry of Special Mappings)
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