Articles of hyperbolic geometry

Existence of unique circle passing through interior points of unit disk meeting the boundary orthogonally

I am a self-studies and this is a hw problem from a complex analysis scourse I’ve been doing. The problem set pertains to the topic Automorphism Groups and has a high concentration of fractional linear transformations. So I would be appreciative of any help, but especially if those concepts are applicable. Show that for any […]

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of $\mathbb{H} / \Gamma$. I do not know why, or what a cusp mathematically is, but I have seen pictures. Then […]

Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?

I read the answers to this very interesting question and saw that we can in fact embed the Euclidean plane into hyperbolic 3-space using what is called a horosphere. However, as Hilbert showed us, the reverse is not true; we cannot embed the hyperbolic plane into Euclidean 3-space. This made me interested in considering the […]

Translating a Euclidean proof to hyperbolic language..

User HyperLuminal asked for help to prove the following statement: Connecting the feet of the altitudes of a given triangle, we obtain another triangle for with the altitudes of the original triangle are angle bisectors of the new triangle. User @Alexey Burdin gave nice answer. Alexey’s answer is based on a fact true only in […]

Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn’t know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then for a given genus $g$, up to isomorphism (holomorphism in our case), there are only finitely many […]

Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?

There is a sense in which all “interesting” properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations correspond to “uninteresting” manipulations of the whole sphere, as illustrated in this video. Is there an equivalent notion in hyperbolic geometry? In other words, is there […]

Models of hyperbolic geometry

Wikipedia states the following: [The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model and the Poincaré disk model (due to Riemann) But it doesn’t give any sources. I would like to know about the real history of the models […]

What's the right way to calculate hyperbolic distance on the hyperboloid model?

I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model: Let $u = (x_0,x_1,x_2)$ and $v = (y_0,y_1,y_2)$ be two points on the positive hyperbolic sheet so $x_0 = 1 + x_1^2 + […]

Hyperbolic area and $SL_2$

Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that: a. The measure $\mu$ is invariant under all $g \in SL_2(\mathbb R)$ i.e. $\mu(gA) = \mu(A)$. b. Compute the hyperbolic area of the standard fundamental domain for the modular group $SL_2(\mathbb Z)$ ( […]

prove that the sum of the angles in any triangle is less than 180 in hyperbolic geometry (or poincare model).

We could use poincare disc model as a hyperbolic geometry model. I have difficulty understanding poincare disc model. So is there someone to help?