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 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 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 On a projectively invariant pseudo-distance in Finsler geometry

2015 ◽  
Vol 12 (04) ◽  
pp. 1550043 ◽  
Author(s):  
Behroz Bidabad ◽  
Maryam Sepasi
Keyword(s):  
Nonlinear Analysis ◽  
Ricci Tensor ◽  
Finsler Geometry ◽  
Finsler Metrics ◽  
Analysis Method ◽  
Covariant Derivatives

Here, a nonlinear analysis method is applied rather than classical one to study projective changes of Finsler metrics. More intuitively, a projectively invariant pseudo-distance is introduced and characterized with respect to the Ricci tensor and its covariant derivatives.

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.

scholarly journals VARIATIONAL FORMULATION OF EISENHART'S UNIFIED THEORY

2009 ◽  
Vol 24 (20n21) ◽  
pp. 3975-3984
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
Variational Formulation ◽  
Ricci Tensor ◽  
Maxwell Equations ◽  
Affine Connection ◽  
Unified Theory ◽  
Field Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field ◽  
Electromagnetic Field Tensor

Eisenhart's classical unified field theory is based on a non-Riemannian affine connection related to the covariant derivative of the electromagnetic field tensor. The sourceless field equations of this theory arise from vanishing of the torsion trace and the symmetrized Ricci tensor. We formulate Eisenhart's theory from the metric-affine variational principle. In this formulation, a Lagrange multiplier constraining the torsion becomes the source for the Maxwell equations.

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.

scholarly journals Dark matter from a dark connection

2021 ◽  
pp. 2150134
Author(s):  
Ranit Das ◽  
Chethan Krishnan
Keyword(s):  
Dark Matter ◽  
Affine Connection ◽  
Vector Model ◽  
Dark Matter Candidate ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Tests Of General Relativity ◽  
Covariant Derivatives ◽  
Composite Field ◽  
Pseudo Scalar

In the first part of this note, we observe that a non-Riemannian piece in the affine connection (a “dark connection”) leads to an algebraically determined, conserved, symmetric 2-tensor in the Einstein field equations that is a natural dark matter candidate. The only other effect it has, is through its coupling to standard model fermions via covariant derivatives. If the local dark matter density is the result of a background classical dark connection, these Yukawa-like mass corrections are minuscule ([Formula: see text] for terrestrial fermions) and none of the tests of general relativity or the equivalence principle are affected. In the second part of the note, we give dynamics to the dark connection and show how it can be re-interpreted in terms of conventional dark matter particles. The simplest way to do this is to treat it as a composite field involving scalars or vectors. The (pseudo-)scalar model naturally has a perturbative shift-symmetry and leads to versions of the Fuzzy Dark Matter (FDM) scenario that has recently become popular (e.g. arXiv:1610.08297) as an alternative to WIMPs. A vector model with a [Formula: see text]-parity falls into the Planckian Interacting Dark Matter (PIDM) paradigm, introduced in arXiv:1511.03278. It is possible to construct versions of these theories that yield the correct relic density, fit with inflation and are falsifiable in the next round of CMB experiments. Our work is an explicit demonstration that the meaningful distinction is not between gravity modification and dark matter, but between theories with extra fields and those without.

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