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Questions tagged [hyperbolic-geometry]

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.

The prototypical example of hyperbolic geometry in two dimensions of Gauss-Lobachevsky-Bolyai in which the parallel postulate of Euclidean geometry is replaced by a new postulate of at least 2 parallel lines through an external point not on the given line with sum of interior angles of a geodesic triangle smaller than $\pi$ radians.

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Matrix representation for generators of the Fuchsian group of the three-holed sphere.

Let $P$ be a three-holed sphere with its usual presentation $$ \pi_1(P) = \langle U, V, W | UVW=1 \rangle. $$ A hyperbolic structure on $P$ is given by a Fuchsian representation $\rho:\pi_1(P) \to \operatorname{PSL}(2, \mathbb{R})$. That is…
user7090
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Equation of a line section in polar hyperbolic space

In the Poincaré representation of the hyperbolic plane a lines is a sections of a circle perpendicular to the unit disk and running through 2 points of interest $x_1$ and $x_2$. I wish to find the analogous curve in the polar hyperbolic…
b sulyok
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Busemann function and its relation to horocycle

Consider Busemann function, $B_{\zeta}(w)$ for the Poincare disk model. Handling with literature, I see that this function relates to the horocycles on the disk. Here (p.148, sec. 9.4.2, def. 9.34) the author just defines horocycle as a level set of…
Artem Alexandrov
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hyperbolic quadrilateral angle

Consider a hyperbolic quadrilateral of $abcd$ in the hyperbolic plane $\mathbb{H}^2$ with the metric being the metric defined via the cayley map. Suppose $\angle b$, $\angle c$ ,$\angle d$ are all of angle $\frac{\pi}{2}$. Show that $ad$ is…
user643073
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How to glue hyperbolic manifolds?

I want to understand the details of gluing for hyperbolic manifolds. By gluing I mean something along the lines of the statement that "a Riemann surface is glued together on the overlap of local charts via holomorphic transition functions" etc. I am…
user7090
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How to define/determine a side-pairing transformation that will generate a Fuchsian group?

From Poincare's Theorem, the side-pairing transformations for a fundamental polygon will generate a Fuchsian group. Existentially and constructively speaking, how am I going to construct or look for a set of side-pairing transformations that will…
Hades Pluto
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Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides?

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides? I have been trying to visually see if that is possible or not.
Richard Carpenter
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Quick Hypberbolic Geometry question concerning Saccheri Quadrilaterals

Can a Saccheri Quadrilateral have 3 congruent sides? I know the summit is less then the base, but could it happen that the base is the same length as the two vertical sides?
Richard Carpenter
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How does my Beltrami-Klein model look?

Did I sketch the picture right based off of the specific instructions given in the problem?
Richard Carpenter
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convex polygons in hyperbolic geometry

Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.
Richard Carpenter
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Hyperbolic geometry (Bolyai)

Bolyai Theorem: Every pair of ulta-parallel lines has a unique line perpendicular to both lines (perpendicular transverse) Does this imply that if there exists a line perpendicular to 2 other lines A and B, then A and B are parallel to each…
user777253
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Solving $R\space \sinh\frac{D}{R}=k$ for $R$

Does a solution exist for $R$ in this equation? I can't seem to solve it either analytically or numerically. $$R\space \sinh\frac{D}{R}=k$$
Quark Soup
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How to construct Fundamental Domains

I understand what a Fundamental Domain is, however, I have a difficulty in understanding the algorithm to construct Fundamental Domain of $SL(2,\mathbb{Z})$. Here are the lecture notes which are quite helpful: Keith Conrad Summer School In these…
Tedebbur
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What induces and what is the meaning of the conformal factor in the Poincaré ball?

I'd like to understand why the conformal factor for the Poincaré ball usually is defined as follows: $$ \lambda_{x}=\frac{2}{1+\kappa\|x\|_2^2} $$ where $\kappa$ is the sectional curvature of the Poincaré ball. I've come across the following…
ndrizza
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the set of all positive (resp. negative) time-like vectors is convex

On page 56 of John Ratcliffe's Foundations of Hyperbolic Manifolds, it says that Corollary 1. The set of all positive (resp. negative) time-like vectors is a convex subset of $\mathbb{R}^n$. I'm confused about this statement. In $\mathbb{R}^{1,…
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