Principal mappings of 3-dimensional Riemannian spaces into spaces of constant curvature

1996 ◽  
Vol 28 (9) ◽  
pp. 1139-1150 ◽  
Author(s):  
Lluís Bel
Keyword(s):  
Constant Curvature ◽  
Spaces Of Constant Curvature ◽  
3 Dimensional ◽  
Riemannian Spaces

scholarly journals Volumes of Polyhedra in Non-Euclidean Spaces of Constant Curvature

2020 ◽  
Vol 66 (4) ◽  
pp. 558-679
Author(s):  
V. A. Krasnov
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Arbitrary Dimension ◽  
Difficult Problem ◽  
Combinatorial Structure ◽  
Euclidean Spaces ◽  
Spaces Of Constant Curvature ◽  
Simple Polyhedron ◽  
3 Dimensional ◽  
Combinatorial Sense

Computation of the volumes of polyhedra is a classical geometry problem known since ancient mathematics and preserving its importance until present time. Deriving volume formulas for 3-dimensional non-Euclidean polyhedra of a given combinatorial type is a very difficult problem. Nowadays, it is fully solved for a tetrahedron, the most simple polyhedron in the combinatorial sense. However, it is well known that for a polyhedron of a special type its volume formula becomes much simpler. This fact was noted by Lobachevsky who found the volume of the so-called ideal tetrahedron in hyperbolic space (all vertices of this tetrahedron are on the absolute).In this survey, we present main results on volumes of arbitrary non-Euclidean tetrahedra and polyhedra of special types (both tetrahedra and polyhedra of more complex combinatorial structure) in 3-dimensional spherical and hyperbolic spaces of constant curvature K = 1 and K = -1, respectively. Moreover, we consider the new method by Sabitov for computation of volumes in hyperbolic space (described by the Poincare model in upper half-space). This method allows one to derive explicit volume formulas for polyhedra of arbitrary dimension in terms of coordinates of vertices. Considering main volume formulas for non-Euclidean polyhedra, we will give proofs (or sketches of proofs) for them. This will help the reader to get an idea of basic methods for computation of volumes of bodies in non-Euclidean spaces of constant curvature.

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.  

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