# Tagged Questions

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

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### Kobayashi distance on the Siegel upper half space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article Intrinsic Metrics on ...
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### Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2}$$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian coordinates....
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### Relationship between regular tessellations in general space

Using the notation for regular tessellations $\{p,q\}$ denoting the tessellation consisting of p-gons , q of which meet at each vertex [e.g. $\{3,6\}$ in $\mathbb{R}^{2}$ is the equilateral triangle ...
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### Incidence axioms for the upper half plane (complex plane).

The axiom I am checking for this question is I1: "For any two distinct points A,B there exist a unique line L containing both points." Show that I1 is satisfied in the upper half of the complex plane ...
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### another pythagorean theorem in hyperbolic geometry

on https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry it says However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition ...
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### What is the radius of the inscribed circle of an ideal triangle

I wanted to calculate the radius of the inscribed circle of an ideal triangle. and when i dat calculate it i came to $\ln( \sqrt {3}) \approx 0.54$ (being arcos(sec (30^o)) but then at https://en....
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### Where are the vertices of the universal cover of the genus 2 torus octagon?

The universal cover of the genus 2 torus is hyperbolic plane and the fundamental domain is a octagon. Here is a picture, which I took from here. Is there a closed form for the points of set of the ...
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### In hyperbolic geometry, prove that parallel lines are not equidistant

In Euclidean Geometry, parallel lines are equidistant. In hyperbolic geometry, it appears that parallel lines are $not$ equidistant. Is there a proof that supports this, or is it supposed to be ...
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### Hyperbolic geometry question concerning lengths between parallel lines

Theorem (H16). If: $l$ and $m$ are parallel lines, $j$ is a common perpendicular intersecting $l$ at point A and $m$ at point B, and C and E are points on $l$ so that C is between A and E, Then: ...
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### Lie groups for beginners: Lie group of hyperbolic geometry

I am trying to understand Lie groups and their relation to (2 dimensional) hyperbolic geometry. as far as I understand it (which is not very far, I am pushing my understanding here) the Lie-group ...
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### spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
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### Volumes of hyperbolic manifolds

In a talk I attended the speaker said that the volume of a closed hyperbolic manifold is a topological invariant. Are known volumes rational? Irrational? Transcendental?
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### Algebraic solutions for Poincaré Disk arcs

Given two points on the Poincaré Disk, there is a single straight line or arc that passes through them and that is orthogonal to the unit circle. Using compass and straightedge methods, one can easily ...
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### “Every geometry is a projective geometry!” So Hyperbolic geometry is a projective geometry?

The great mathematician Arthur Cayley (https://en.wikipedia.org/wiki/Arthur_Cayley ) seems to have said "all geometry is projective geometry" (sorry no exact source, probably it is somewhere in Felix ...
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### Why does the halfplane model of the hyperbolic plane involve only the upper half of the plane?

The Hyperbolic metric $$s= \pm \int \frac {\sqrt{1 +y'^2} \, dx}{y}$$ for geodesics in $\mathbb H^2$ integrates to a full circle, but why only the upper half is considered? The query is about ...
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### Applications of the Hurwitz Theorem on Number of Automorphisms?

So I am giving a talk in which we'll prove the semi-famous Hurwitz Theorem: Let $S$ be a compact Riemann surface of genus $g \geq 2$. Then $|Aut(S)| \leq 84(g-1)$, the group of holomorphic ...
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### 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 ...
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### Outer Automorphisms of PSL2(R)

As far as I've been able to tell, a description of Out$(PSL_2(\mathbb{R}))$ isn't available online. I also looked in Lang's $SL_2(\mathbb{R})$ but it's not discussed. I guess my first question would ...
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### Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic 3-space. Let $TH$ be the tangent bundle of $H$. I have a question: Is $TH$ isometric to H times a flat $k$-space?
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### Triangle inequality of hyperbolic metric

For $z_1, z_2 \in \mathbb{B}^2$, define $d(z_1, z_2) = \text{cosh}^{-1}(1+ \dfrac{2|z_1 - z_2|^2}{(1-|z_1|^2)(1-|z_2|^2)})$. In my text book (Lee's Topological manifolds Problems 12-23), to prove ...
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### Do minimal hyperbolic surfaces exist? What do they look like?

I understand that it is impossible to embed* the entire hyperbolic plane in $\mathbb{R}^3$. But, can one create a embedding of part of the hyperbolic plane such that the resulting surface is also ...
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### What about the Fuchsian groups make them stand out?

Why do we stop at Fuchsian groups (I.e. discrete subgroups of automorphisms of the hyperbolic plane) when we study things like quotients and what not? Is there a maximalist or universality property ...
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### Is it true that a geodesic on a hyperbolic surface can be lifted to a geodesic on the hyperbolic plane?

Let $\mathbb{H}$ be the hyperbolic plane, $\Gamma < \text{Isom}(\mathbb{H})$ be a Fuchsian group, and $S = \mathbb{H}/\Gamma$. If $\gamma : [0,1] \rightarrow S$ is a geodesic, can it be lifted to a ...
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### I am looking for an introduction to hyperbolic surfaces as a quotient of the upper half plane by lattices.

I keep coming across results of the form: If we take the quotient of the upper half plane by a Fuchsian group with this property, we get a surface with that property (cusps, funnels, in/finite area, .....