Optimal regularity of constant curvature graphs in Hyperbolic space

2019 ◽  
Vol 58 (4) ◽  
Author(s):  
Xumin Jiang ◽  
Ling Xiao
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Optimal Regularity

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 Hypersurfaces of Constant Curvature in Hyperbolic Space I

2009 ◽  
Vol 19 (4) ◽  
pp. 772-795 ◽  
Author(s):  
Bo Guan ◽  
Joel Spruck ◽  
Marek Szapiel
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature

Non-Euclidean Extensions

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Positive Curvature ◽  
Spherical Geometry ◽  
Two Dimensional ◽  
Actual Infinity ◽  
Constant Positive Curvature ◽  
Negative Constant ◽  
The Given ◽  
Line P

This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.

Minimal rotational surfaces in the product space ℚ𝜀2 × 𝕊1

2018 ◽  
Vol 29 (08) ◽  
pp. 1850051
Author(s):  
Güler Gürpınar Arsan ◽  
Uğur Dursun
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Product Space ◽  
Inner Product ◽  
Rotational Axis ◽  
Rotational Surfaces ◽  
Unit Formula

In this work, we study minimal rotational surfaces in the product space [Formula: see text], where [Formula: see text] denotes either the unit [Formula: see text]-sphere [Formula: see text] or the [Formula: see text]-dimensional hyperbolic space [Formula: see text] of constant curvature [Formula: see text], according to [Formula: see text] or [Formula: see text], respectively. While there is only one kind of rotational surfaces in [Formula: see text], there are three different possibilities for rotational surfaces in [Formula: see text], according to the types of the induced inner product on the rotational axis of the surface. We determine the profile curves of all minimal rotational surfaces in [Formula: see text].

Corresponding Polyhedra in the Three Spaces of Constant Curvature

1953 ◽  
Vol 5 ◽  
pp. 40-45
Author(s):  
Ernst Roeser
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Constant Curvature ◽  
Dihedral Angles ◽  
Euclidean Spaces ◽  
Platonic Solids ◽  
Spaces Of Constant Curvature ◽  
Elliptic Space ◽  
The Face

The five Platonic solids can be drawn in elliptic or hyperbolic space just as well as in Euclidean space. Their numerical properties are, of course, the same in all three. So are the various angles subtended at the centre. But the face-angles and dihedral angles are greater in elliptic space, smaller in hyperbolic. It is a special feature of the non-Euclidean spaces that we cannot change the size of a solid without changing its shape.

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.

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