Separation of variables for complex Riemannian spaces of constant curvature - I. Orthogonal separable coordinates for S n c and E n c

1984 ◽  
Vol 394 (1806) ◽  
pp. 183-206 ◽  
Keyword(s):  
Separation Of Variables ◽  
Complete Solution ◽  
Complete Classification ◽  
Conformally Flat ◽  
Coordinate Systems ◽  
Spaces Of Constant Curvature ◽  
Orthogonal Coordinate Systems ◽  
Hamilton Jacobi Equation ◽  
Riemannian Spaces

A complete classification of all orthogonal coordinate systems that admit a separation of variables for the null Hamilton-Jacobi equation in conformally flat complex Riemannian spaces is presented. This is a first step towards the complete solution of the problem for complex Riemannian spaces when, in general, the coordinates need not be orthogonal. A detailed prescription for constructing all such orthogonal coordinate systems is presented.

scholarly journals Shapovalov Wave-Like Spacetimes

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1372 ◽  
Author(s):  
Konstantin Osetrin ◽  
Evgeny Osetrin
Keyword(s):  
Jacobi Equation ◽  
Eikonal Equation ◽  
Separation Of Variables ◽  
Complete Classification ◽  
Complete Separation ◽  
Einstein Equations ◽  
Coordinate Systems ◽  
Test Particles ◽  
Hamilton Jacobi Equation

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

The complete classification of a class of conformally flat Lorentzian hypersurfaces in ℝ14

2017 ◽  
Vol 28 (13) ◽  
pp. 1750092
Author(s):  
Zhenxiao Xie ◽  
Changping Wang ◽  
Xiaozhen Wang
Keyword(s):  
Three Dimensional ◽  
Complete Classification ◽  
Conformally Flat ◽  
Space Forms ◽  
Lorentzian Space ◽  
Lorentzian Metric ◽  
Cone Model ◽  
Shape Operators ◽  
Frenet Curves

A three-dimensional Lorentzian hypersurface [Formula: see text] is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of [Formula: see text]. In this paper, using the projective light-cone model, we give a complete classification of those ones whose shape operators have two distinct real eigenvalues and cannot be diagonalizable. These hypersurfaces are conformal equivalent to cones, cylinders, or rotational hypersurfaces generated by B-scrolls (over null Frenet curves) in three-dimensional Lorentzian space forms.

scholarly journals On conformally reducible pseudo-Riemannian spaces

2021 ◽  
Vol 14 (2) ◽  
pp. 154-163
Author(s):  
Тетяна Iванiвна Шевченко ◽  
Тетяна Сергіївна Спічак ◽  
Дмитро Миколайович Дойков
Keyword(s):  
General Theory ◽  
Conformal Mapping ◽  
Main Type ◽  
Complete Classification ◽  
Riemann Tensor ◽  
Conformally Flat ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Einstein Spaces

The present paper studies the main type of conformal reducible conformally flat spaces. We prove that these spaces are subprojective spaces of Kagan, while Riemann tensor is defined by a vector defining the conformal mapping. This allows to carry out the complete classification of these spaces. The obtained results can be effectively applied in further research in mechanics, geometry, and general theory of relativity. Under certain conditions the obtained equations describe the state of an ideal fluid and represent quasi-Einstein spaces. Research is carried out locally in tensor shape.

scholarly journals Special semi-reducible pseudo-Riemannian spaces

2021 ◽  
Vol 14 (1) ◽  
pp. 48-59
Author(s):  
Юлія Степанівна Федченко ◽  
Олександр Васильович Лесечко
Keyword(s):  
Constant Curvature ◽  
Symmetric Spaces ◽  
Necessary Conditions ◽  
Conformally Flat ◽  
Spaces Of Constant Curvature ◽  
Riemannian Spaces

The paper contains necessary conditions allowing to reduce matrix tensors of pseudo-Riemannian spaces to special forms called semi-reducible, under assumption that the tensor defining tensor characteristic of semireducibility spaces, is idempotent. The tensor characteristic is reduced to the spaces of constant curvature, Ricci-symmetric spaces and conformally flat pseudo-Riemannian spaces. The obtained results can be applied for construction of examples of spaces belonging to special types of pseudo-Riemannian spaces. The research is carried out locally in tensor shape, without limitations imposed on a sign of a metric.  

scholarly journals The spacetime models with dust matter that admit separation of variables in Hamilton–Jacobi equations of a test particle

2016 ◽  
Vol 31 (06) ◽  
pp. 1650027 ◽  
Author(s):  
Konstantin Osetrin ◽  
Altair Filippov ◽  
Evgeny Osetrin
Keyword(s):  
Velocity Field ◽  
Energy Density ◽  
Test Particle ◽  
Jacobi Equation ◽  
Functional Form ◽  
Separation Of Variables ◽  
Complete Separation ◽  
Coordinate Systems ◽  
Jacobi Equations ◽  
Hamilton Jacobi Equation

The characteristics of dust matter in spacetime models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton–Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.

Curvature models of conformally flat Walker (2,2)-manifolds

2019 ◽  
Vol 16 (05) ◽  
pp. 1930002
Author(s):  
Mohamad Chaichi
Keyword(s):  
Curvature Tensor ◽  
Complete Classification ◽  
Conformally Flat ◽  
Signature Formula

Four-dimensional conformally flat curvature models of signature [Formula: see text] are considered and complete classification of curvature tensor is obtained. Then, some remarks about Ivanov–Petrova and Walker–Ivanov–Petrova properties are stated.

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