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

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1560
Author(s):  
Volodymyr Berezovski ◽  
Yevhen Cherevko ◽  
Irena Hinterleitner ◽  
Patrik Peška
Keyword(s):  
Symmetric Spaces ◽  
Affine Connection ◽  
Cauchy Type ◽  
Theory Of Relativity ◽  
Covariant Form ◽  
General Theory Of Relativity ◽  
Nontrivial Solutions ◽  
Affine Connections ◽  
Systems Of Pdes ◽  
Mixed Systems

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≥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.

Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connection Onto Generalized 2-Ricci-Symmetric Spaces

2021 ◽  
Vol 22 ◽  
pp. 78-87
Author(s):  
Volodymyr Berezovski ◽  
Yevhen Cherevko ◽  
Svitlana Leshchenko ◽  
Josef Mikes
Keyword(s):  
Differential Equations ◽  
Closed System ◽  
Symmetric Spaces ◽  
Affine Connection ◽  
Linear Differential Equations ◽  
Cauchy Type ◽  
Affine Connections ◽  
Covariant Derivatives ◽  
Linear Differential

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces. The main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained result extends an amount of research produced by Sinyukov, Berezovski and Mike\v{s}.

scholarly journals Conformal and Geodesic Mappings onto Some Special Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 664 ◽  
Author(s):  
Volodymyr Berezovski ◽  
Yevhen Cherevko ◽  
Lenka Rýparová
Keyword(s):  
Differential Equations ◽  
Closed System ◽  
Symmetric Spaces ◽  
Conformal Mappings ◽  
Cauchy Type ◽  
Affine Connections ◽  
Covariant Derivatives ◽  
Riemannian Spaces

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.

Tensors

2021 ◽  
pp. 41-77
Author(s):  
Moataz H. Emam
Keyword(s):  
General Theory ◽  
Parallel Transport ◽  
Three Dimensions ◽  
Theory Of Relativity ◽  
Covariant Form ◽  
General Theory Of Relativity ◽  
Coordinate Systems ◽  
Christoffel Symbols ◽  
Metric Connection

In this chapter we develop the concept of tensors, their meaning, and how they arise from vectors. Emphasis is placed on tensor transformations, covariance between coordinate systems, and relation to the metric. The concept of metric connection and the Christoffel symbols is introduced in three dimensions via the easily visualizable idea of parallel transport. Derivatives and intergrals in covariant form are discussed. The first two chapters are designed to familiarize the reader with the language that is the bread and butter of the general theory of relativity and other higher geometric theories.

scholarly journals On Canonical Almost Geodesic Mappings of Type π2(e)

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 54 ◽  
Author(s):  
Volodymyr Berezovski ◽  
Josef Mikeš ◽  
Lenka Rýparová ◽  
Almazbek Sabykanov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symmetric Spaces ◽  
Riemann Tensor ◽  
Mixed System ◽  
Cauchy Type ◽  
Partial Differential ◽  
Affine Connections

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.

scholarly journals On canonical F-planar mappings of spaces with affine connection

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1273-1278
Author(s):  
Volodymyr Berezovski ◽  
Josef Mikes ◽  
Patrik Peska ◽  
Lenka Rýparová
Keyword(s):  
General Solution ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Affine Connection ◽  
Cauchy Type ◽  
Partial Differential ◽  
Covariant Derivatives ◽  
Type Form

In this paper we study the theory of F-planar mappings of spaces with affine connection. We obtained condition, which preserved the curvature tensor. We also studied canonical F-planar mappings of space with affine connection onto symmetric spaces. In this case, the main equations have the partial differential Cauchy type form in covariant derivatives. We got the set of substantial real parameters on which depends the general solution of that PDE?s system.

scholarly journals Geodesic mappings of spaces with affine connnection onto generalized Ricci symmetric spaces

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4475-4480 ◽  
Author(s):  
V.E. Berezovski ◽  
Josef Mikes ◽  
Lenka Rýparová
Keyword(s):  
Symmetric Spaces ◽  
Fundamental System ◽  
Affine Connection ◽  
Cauchy Type ◽  
Covariant Derivatives

The presented work is devoted to study of the geodesic mappings of spaces with affine connection onto generalized Ricci symmetric spaces. We obtained a fundamental system for this problem in a form of a system of Cauchy type equations in covariant derivatives depending on no more than 1/2 n2(n+1)+n real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto equiaffine generalized Ricci symmetric spaces.

scholarly journals Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 437
Author(s):  
Volodymyr Berezovski ◽  
Yevhen Cherevko ◽  
Josef Mikeš ◽  
Lenka Rýparová
Keyword(s):  
Differential Equations ◽  
Closed System ◽  
Symmetric Spaces ◽  
Linear Differential Equations ◽  
Cauchy Type ◽  
Affine Connections ◽  
Covariant Derivatives ◽  
Linear Differential

In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš.

The general theory of relativity is correct!

1988 ◽  
Vol 155 (7) ◽  
pp. 517-527 ◽  
Author(s):  
Ya.B. Zel'dovich ◽  
Leonid P. Grishchuk
Keyword(s):  
General Theory ◽  
Theory Of Relativity ◽  
General Theory Of Relativity

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

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