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 Geodesic mappings of manifolds with affine connection onto the Ricci symmetric manifolds

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 379-385 ◽  
Author(s):  
Volodymyr Berezovskii ◽  
Irena Hinterleitner ◽  
Josef Mikes
Keyword(s):  
Ricci Tensor ◽  
Fundamental System ◽  
Affine Connection ◽  
Cauchy Type ◽  
Covariant Derivatives

In the present paper we investigate geodesic mappings of manifolds with affine connection onto Ricci symmetric manifolds which are characterized by the covariantly constant Ricci tensor. 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 n(n+1) real parameters. Analogous results are obtained for geodesic mappings of manifolds with affine connection onto symmetric manifolds.

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.

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.

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.

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.

scholarly journals On higher covariant derivatives of the curvature tensors of Kählerian C-spaces

1983 ◽  
Vol 91 ◽  
pp. 1-18 ◽  
Author(s):  
Ryoichi Takagi
Keyword(s):  
Symmetric Spaces ◽  
Compact Type ◽  
Hermitian Symmetric Spaces ◽  
Covariant Derivatives ◽  
Higher Covariant Derivatives ◽  
Group Of Isometries ◽  
Complex Homogeneous Manifold ◽  
Curvature Tensors ◽  
Simply Connected ◽  
Derivatives Of

A compact simply connected complex homogeneous manifold is said briefly a C-space, which was completely classified by H. C. Wang [12]. A C-space is called to be Kählerian if it admits a Kählerian metric such that a group of isometries acts transitively on it. Hermitian symmetric spaces of compact type are typical examples of a Kählerian C-space. Let M be an arbitrary Kählerian C-space and R its curvature tensor. M. Itoh [6] expressed R in the language of Lie algebra and investigated various properties of R. In this paper, we study higher covariant derivatives of R.

scholarly journals Divergent part of the stress-energy tensor for Maxwell’s theory in curved space-time: a systematic derivation

2021 ◽  
Vol 136 (5) ◽  
Author(s):  
Roberto Niardi ◽  
Giampiero Esposito ◽  
Francesco Tramontano
Keyword(s):  
Green Function ◽  
Symmetric Spaces ◽  
Splitting Method ◽  
Space Time ◽  
Energy Tensor ◽  
Divergent Part ◽  
Stress Energy Tensor ◽  
Curved Space ◽  
Covariant Derivatives ◽  
Stress Energy

AbstractIn this paper the Feynman Green function for Maxwell’s theory in curved space-time is studied by using the Fock–Schwinger–DeWitt asymptotic expansion; the point-splitting method is then applied, since it is a valuable tool for regularizing divergent observables. Among these, the stress-energy tensor is expressed in terms of second covariant derivatives of the Hadamard Green function, which is also closely linked to the effective action; therefore one obtains a series expansion for the stress-energy tensor. Its divergent part can be isolated, and a concise formula is here obtained: by dimensional analysis and combinatorics, there are two kinds of terms: quadratic in curvature tensors (Riemann, Ricci tensors and scalar curvature) and linear in their second covariant derivatives. This formula holds for every space-time metric; it is made even more explicit in the physically relevant particular cases of Ricci-flat and maximally symmetric spaces, and fully evaluated for some examples of physical interest: Kerr and Schwarzschild metrics and de Sitter space-time.

Conditions for the local existence of metric in a generic affine manifold

1980 ◽  
Vol 87 (3) ◽  
pp. 527-534 ◽  
Author(s):  
Kuo-Shung Cheng ◽  
Wei-Tou Ni
Keyword(s):  
Affine Connection ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Riemann Tensor ◽  
Affine Manifold ◽  
Absolute Value ◽  
First Order ◽  
Covariant Derivatives ◽  
Explicit Formulae ◽  
Necessary And Sufficient

AbstractFor a manifold with a generic symmetric affine connection, explicit necessary and sufficient conditions for the local existence of metric compatible with the connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives. If these conditions are satisfied, the solutions for metric are unique up to a constant scale factor and the absolute value of the signature is uniquely determined. Explicit formulae for the solutions are given in terms of integrals.

Conditions for an affine manifold with torsion to have a Riemann–Cartan structure

1981 ◽  
Vol 90 (3) ◽  
pp. 517-527 ◽  
Author(s):  
Wei-Tou Ni
Keyword(s):  
Affine Connection ◽  
Local Existence ◽  
Sufficient Conditions ◽  
Riemann Tensor ◽  
Scale Factor ◽  
Necessary And Sufficient Conditions ◽  
Affine Manifold ◽  
First Order ◽  
Covariant Derivatives ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions for the local existence of a metric compatible with the affine connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives in a generic affine manifold with torsion. In case these conditions are satisfied, the solutions of the metric are given in terms of integrals and are unique up to a constant scale factor. Some global conditions are also obtained and discussed.

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