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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Conformal Map from Intersection of Two Discs and Half-Plane

I have one of those "find the map" problems that is really giving me a lot of trouble. Let $B_1(1)$ be the ball of radius $1$ centered at $1$. We have the following domain: $\mathbb{D} \cap B_1(1) \...
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How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
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Isoperimetric inequality in the Poincare disc

In the Poincare disc model, a version of the isoperimetric inequality states that $L(\partial(A))>c\mu(A),$ where $\mu$ is the hyperbolic area and $L(\gamma)=\int_0^1 \frac{|\gamma'(t)|}{1-|\gamma(...
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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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Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ...
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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 $...
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Prove $A \cdot B\leq -1$, where $A$ and $B$ are in $\mathbb{H}^2$

Let $A$ and $B$ be in $\mathbb{H}^2$. I need to prove that the lorentzian dot product between $A$ and $B$ is less than or equal to $-1$. I have no idea where to start.
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Non euclidean lines (finding endpoints of semicircle)

A non euclidean line in $\mathbb{RP}^1$ in terms of reflections about the unit circle can be written in the form $A+B(\overline{w}+w)+C(\overline{w}w)=0$ Where $w=\frac{1}{\overline{z}}$ The ...
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sympletic form of the hyperboloid

I'm working with the Hyperboloid of two leefs $H_2$ as a coadjoint orbit of $\mathfrak{sl}^*(2, R)$. I know that $H_2$ is a symplectic manifold by the following theorem: Given a Lie group G and $\...
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Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...
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Villarceau circle as a Loxodrome

A circular Clifford torus (radius at flat circle = h, section radius $ a , a<h $ ) is cut by a plane at an angle $ \cos \alpha = a/h \tag{1} $ centrally to the symmetry axis, the line of ...
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Example of a doubly degenerate Kleinian group which does not come from a mapping torus

Doubly degenerate Kleinian groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as follows:...
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Can a hyperbolic surface be isometrically embedded into $\mathbb R^4$?

Can a complete hyperbolic surface be isometrically embedded into flat $\mathbb R^4$?
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When a ray of an horocircle passing through the origin intersects the y axis.

In the following figure, $h(A,B)$ is an horocycle centered in A passing over B. $\Theta(h)$ is the angle of parallelism of the segment $h$ and $S$ is the well known intersection of a chord of an ...
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Tiling on Poincaré disc [closed]

Is there anyone to help me tile on a Poincaré disc? In fact, I'm going to tile triangle tiles on a surface in hyperbolic geometry ; is there any algorithmic method to do so?
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Mapping the Poincaré disk to hyperbolic surfaces in $\mathbb{R}^3$.

Take any hyperbolic surface with constant curvature in $\mathbb{R}^3$, such as Dini's surface, or a hyperboloid of constant curvature. If I understood things correctly, for any such surface, we ...
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How to make a triangular pillow with a pre-drilled tube with two truncated tetrahedra? (from Jeff Weeks' paper)

In his paper Computation of Hyperbolic Structures in Knot Theory, p.12, Jeff Weeks explains as below how to make a triangular pillow with a pre-drilled tube by gluing two truncated tetrahedra. A ...
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59 views

Compute of curvature

In the answer of this question,for the given metric $$g_M = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2,$$ how to compute the curvature? Whether the hyperbolic space means $M=\{x\in R^n:x_n>0\}$? ...
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An $d$-unramified covering of compact Riemann surfaces induce a (monodromy) action on $d$ letters. Is the opposite true?

Let $S_1, S$ be compact connected Riemann surfaces, $f : S_1 \rightarrow S$ be a meromorphic function of degree $d$ that branch over $B \subset S$. The unmarried covering $f : S_1 \backslash f^{-1}(B) ...
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Hyperbolic space and metrics

Using metrics is it possible to derive the circumference and area of a circle in hyperbolic space. I've found that the answer (without using metrics) are: C=2πsinh(r) and A=4πsinh2(r/2). But I'm ...
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Constant of a hyperbola

Hyperbolas are a companion to a circle, sharing many properties when it comes to their trig functions and equation. But, if the circle has $\pi$ as a constant relation, does a hyperbola have some ...
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How to show that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures?

I am reading the lecture notes. On page 19, let $G=SL_2$. It is said that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures on the ...
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Is the fundamental group of a compact Riemann surface *after* removing a finite number of points still a Fuchsian group?

Let $S$ be a compact R.S. admitting a Fuchsian model $\mathbb{H} / \Gamma$. We know that $\pi_1(S) \cong \Gamma$. Let $\mathcal{B} \subseteq S$ be a finite set of points, is $\pi_1(S - \mathcal{B})$ ...
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Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
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The precise formula of the Poincare-Bergman metric on the disc $\mathbb{D}$.

Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. If $\gamma:[0,1]\rightarrow{\mathbb{D}}$ is a $C^1$ curve in $\mathbb{D}$, we define the Bergman length of $\gamma$ by $$l_B(\gamma)=\int_0^1\frac{|\...
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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, .....
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169 views

Area of a right angled hyperbolic triangle as function of side lengths

I was puzzeling with Area of hyperbolic triangle definition and could not figure it out, but then i thought there should be a (maybe solvable) simpler problem so here it is: suppose: an ...
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Various definitions of Moduli space of Riemann surfaces and Uniformization theorem

I'm sorry for the quantity of questions I'm asking, but I would like to solve once and for all many doubts I have on equivalent definitions of the moduli space of Riemann surfaces. Definition 1: The ...