# Questions tagged [hyperbolic-geometry]

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.

1,193 questions
34 views

### Universal cover of a not-necessarily-complete hyperbolic manifold

The Cartan-Hadamard theorem (as typically stated) tells us that the universal cover of a geodesically complete and connected Riemannian manifold $M$ with non-positive sectional curvature is ...
48 views

### Simplification of $\sin(\pi^x)$ , with $x$ being a positive irrational number

How to simplify if $a > 0$ and $\cos(a) < 0$ Was a previous post. Correction, it was suppose to be if a > 0 & cos(a) > 0. An answer was given. https://math.stackexchange.com/a/1274372/...
13 views

### non-uniformly hyperbolic definition

I'm working on Uniformly hyperbolic finite-valued $SL(2,R)$ -cocycles( article from Arthur Aveila and Jairo Bochi)and at the beginning of my researches,i want to know the exactly meaning of the title ...
17 views

### Hyperbolic isometry and line segments

I was trying to apply Poincare's Polygon theorem, for that I had to give a pairing of sides, i.e., to have an isometry of the hyperbolic plane that will take a side of a polygon to another side (of ...
20 views

### How to prove the existence of infinite geodesics that do not intersect in hyperbolic space

My given question is: In hyperbolic space, given a geodesic $L$ and a point $p$ not lying on $L$, show there is an infinite number of geodesics through $p$ which do not intersect $L$. The "model" ...
707 views

### Magnifying glass in hyperbolic space

My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such ...
13 views

### Right triangle circumscribed by a horocycle

Is it possible that a right triangle is circumscribed by a horocycle? Or, is this statement a theorem in the hyperbolic gemetry? For any horocycle $\gamma$, there are no three distinct ordinary ...
21 views

### Hyperbolic 3-manifolds of finite volume as link complements

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference). Let $N$ be a hyperbolic 3-manifold of finite volume ...
43 views

### Can this equation $b^2$ = $c^2-a^2$ be derived intuitively?

Today while proving the equation of hyperbolas,$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\tag{1}$$ I came across this expression $$b^2=c^2-a^2\tag 2$$ Though this expression seems much like Pythagorean ...
39 views

### Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (...
25 views

### Is every hyperbolic 3-manifold of finite volume a link complement in some closed 3-manifold?

The question says all I need to know, but I will try to clarify it a little more. Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic ...
20 views

### Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space? Is ...
43 views

42 views

### Complex-analytic isometry of hyperbolic ball and half-space

I'm trying to prove that the $n$-dimensional Poincaré ball and half-space models of hyperbolic space are isometric. Here the Poincaré ball $\mathbb B^n_R$ ($n$-ball of radius $R$) has the Riemannian ...
23 views

### Riemann Mapping theorem in triangulations

I am reading the paper 'Rotation Distance, Triangulations, and Hyperbolic Geometry' by Thurston et al. The authors are constructing a sequence of triangulation from a regular icosahedron. Each face of ...
25 views

47 views

### Finding the point through which the tangent passes

A hyperbola passes through the point $P=(\sqrt{2},\sqrt{3})$ and has foci $(\pm 2,0)$. Show that the tangent at $P$ passes through the point $(2\sqrt{2},3\sqrt{3})$. Attempt: So the equation of ...
22 views

### geodesics in hyperbolic space

Let $M$ be the Poincare ball model of the Hyperbolic space, and let $\zeta \in T_0M$. In my lecture notes it is claimed that $$c(t)=\tanh(\Vert \zeta \Vert t )\zeta/\Vert \zeta \Vert$$ is the geodesic ...
18 views

### a good description of the Cayley--Klein models especially about its homogeneity property

Actuality, I'm working with conic in hyperbolic geometry and I'M looking for a good description of the Cayley--Klein models especially about its homogeneity property?
61 views

46 views

### Differential Action of Möbius Transformations

The group $\mathrm{PSL}_2(\mathbb{R})$ acts on $\mathbb{H}$ via Möbius transformations, that is \begin{align*} g=\begin{pmatrix} a & b \\ c & d\end{pmatrix}:z\mapsto \frac{az+b}{cz+d}. \end{...
52 views

### Description of Model of Euclidean Geometry found in the Hyperbolic Plane

I have read that there is a model of Euclidean Geometry in the Hyperbolic Plane, but can't find any description on the web in a digestible form and thought I'd ask this question: If one can describe ...
16 views

22 views