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

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hyperbolic geometry proof with parallel lines

We are assuming hyperbolic geometry in this proof. Prove that for every line $l$ and external point P (im assuming point $P$ is not on line $l$), there are an infinite number of distinct lines ...
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40 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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1answer
18 views

Showing reflections are hyperbolic isometries in $\mathbb{D}$.

I am interested in showing that isometries in $\mathbb{D}$ are either conformal self-maps in $\mathbb{D}$ or they are compositions of conformal self-maps with $z\mapsto \bar{z}$. It is given that ...
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1answer
21 views

Invariance of measure on upper half plane

The upper half plane has the measure $|y|^{-2}dxdy$. Show that it is invariant under the action of $SL(2, \mathbb{R})$. I don't understand what any of this means. First, I don't understand what they ...
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1answer
8 views

Hyperbolic inversions are transitive on unit vectors at $x \in D$

Consider the Poincaré model in which the hyperbolic plane is the interior of a disk $D$, and a point $x$ in it with two vectors $v$ and $w$ of the same length attached. The reflection with respect to ...
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1answer
48 views

How to construct a triangle with divergently parallel perpendicular bisectors?

I'm pretty sure it is possible to construct a triangle in the Klein model of hyperbolic geometry such that the perpendicular bisectors are divergently parallel, but I'm struggling to do so. I've been ...
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2answers
48 views

conformal map/Mobius transformation from annulus to $\mathbb{C}\setminus \overline{D(0,1)}$

Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? Does there exist a conformal bijection/Mobius transformation from the annulus $\{z\in ...
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1answer
23 views

Horocycles in $\mathbb{H}$ and how to measure the distance between them.

Suppose we are looking at the hyperbolic plane $\mathbb{H}$ with usual metric. Now let $u,v \in \mathbb{R} (u < v)$ and consider the unique geodesic joining them. Now consider horocylces at both $u ...
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41 views

Why does $\mathrm{Aut}(\mathbb{H}) = \mathrm{Isom}^{+}(\mathbb{H})$?

Suppose $T \in \mathrm{Isom}^{+}(\mathbb{H})$ . With out loss of generality we may assume that $T$ fixes two points $P,P′$ on the imaginary axis $i\mathbb{R}$. Now let $Q \in \mathbb{H}$. Since ...
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1answer
23 views

Triangular tiling of hyperbolic plane

I'm currently trying to read an interesting paper having to do with embedding graphs in hyperbolic spaces. Namely, "Geographic Routing Using Hyperbolic Space" by Robert Kleinberg. Link: ...
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20 views

Big picture question about the Thurston Metric on Teichmuller Space

I'm having difficulty in appreciating the significance of the Thurston metric on Teichmuller space. It seems like a lot of work to develop the theory of this asymmetric metric space with fewer ...
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1answer
82 views

Prove that the Euclid's parallel postulate is false on the hyperboloid model

Consider the two-dimensional case, i.e. of the hyperbolic place. Define our hyperboloid as the set of points $x=(x_1,x_2,x_3)$ in 3-space (note: Minkowski space, but not needed for this problem) that ...
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0answers
11 views

Finding a fundamental polygon for two-generator subgroup of PSL(2,R)

Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like ...
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1answer
35 views

What is the most accurate definition of the hyperboloid model of hyperbolic geometry?

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a ...
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1answer
23 views

What is PSU(1,1) PSU(2,1)? Is there a general group of the form PSU(n,m)?

I have seen it written about groups PSU(1,1), PSU(2,1). But what exactly are these? The definitions were not given, and I can't seem to find a definition online. Moreover is there a general class of ...
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2answers
51 views

What does it mean when someone writes $ds = \frac{1}{y}(dx^2+dy^2)$?

What does it mean when someone writes $ds = \frac{1}{y}(dx^2+dy^2)$? (I have seen this is written in the setting of hyperbolic space.) But essentially I have no idea as to how to interpret this ...
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1answer
69 views

Defining a Hyperbolic Metric in a General Surface S

I hope someone can help me or give a ref. I'm trying to understand the general way in which one defines a hyperbolic metric on a given surface $\Sigma$ ( and, if not too complicated, on a manifold of ...
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38 views

Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies ...
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1answer
54 views

Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
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0answers
105 views

Area in the upper half plane of a domain-Collar lemma

How can I compute the area of this region in the upper half plane $N=\{z\in U| 0<|z|<e^a,|\text{arg}(z)-\frac{\pi}{2}|<\theta_0, \text{Re}(z)\ge 0\}$, where $U$ denotes the upper half plane ...
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1answer
135 views

Constructing a regular right angled hyperbolic hexagon

I would like to construct a regular right angled hexagon in a klein model. I'm having a hard time understanding why this method works, here is what my professor did in class. Any additional comments ...
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1answer
99 views

hyperbolic geometry (and circle ) construction problem

Was thinking about hyperbolic geometry, the Poincare Disk Model and Sweikarts constant and combined them all in a construction puzzle that I was unable to solve. My construction puzzle: Given: A ...
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2answers
37 views

Is the hyperbolic plane the only simply connected hyperbolic 2-manifold?

Let $S$ be a simply connected Riemannian 2-manifold with everywhere negative curvature. Is $S$ necessarily diffeomorphic to $\mathbb{R}^2$?
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Does a lattice in $PSL(2,\mathbb{R})$ stabilizing $\infty$ have a domain with vertex at $\infty$?

Suppose $\Gamma$ is a lattice in $PSL(2, \mathbb{R})$ acting on the upper half plane. Suppose that the stabilizer in $\Gamma$ of the point at infinity is nontrivial. Does it then follow that the ...
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2answers
66 views

Circle-Circle intersection area in the hyperbolic space

Is there any closed formula for the area of the intersection of two circles in the hyperbolic plane $\mathbb{H}^2$? The two circles have radii $R, R'$ and a distance of $d$ between centers. If ...
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1answer
36 views

Using the Gromov product in inappropriate ways

The Gromov product $(x,y)_z=1/2(d(z,x)+d(z,y)-d(y,x)$ is used in Gromov hyperbolic groups to measure how long two rays stay together or how thin a triangle is. In particular, if $(x,y)_z=n$ in a ...
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1answer
33 views

hyperbolic equilateral triangle : $\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = \frac{1}{2}$

I met this problem in Ratcliffe's Foundations of Hyperbolic Manifolds. Please help me prove this. In an equilateral triangle with side length $a$ and angle $\alpha$, $$\cosh \left(\frac{1}{2} ...
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0answers
45 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
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0answers
38 views

References for Hyperbolic Graph Theory

I'm sorry to disturb you but I really got stuck! I can't find any clear and, somewhat, complete reference for this topic. I'm looking for a book, or review, or survey or course notes regarding ...
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2answers
86 views

Relation: Modular Forms and hyperbolic geometry, or, why do they map from $\mathbb{H}$?

In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic ...
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1answer
57 views

An axiomatic treatment of hyperbolic trigonometry?

I would like to see results derived in hyperbolic trigonometry synthetically, i.e. just by working from axioms, for example the ones given by Hilbert (or even Tarski). Most authors seem to discuss ...
3
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1answer
44 views

Hyperbolic Metric with respect to $\pi: \mathbb{H} \rightarrow {{\Delta}^{*}}$

I have the following problem: Find the unique metric $\rho=\rho(z)\left|dz\right|$ on the punctured unit disk $\Delta^{*}$ such that $\pi^{*}(\rho)=\left|dz\right|/(\mathrm{Im}(z))$ where $\pi: ...
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2answers
56 views

What is the distance from the origin to a right angled regular hyperbolic octagon?

Given a right angled regular hyperbolic octagon centered at origin, what is the distance from the origin to any vertex? I know that the distance between the origin and the point $p=(a,0)$, $a>0$, ...
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2answers
206 views

How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?

This is in response to comments and the answer by user studiosus to this question: As for Beltrami's work: Consistency of a geometry from (post) Hilbert viewpoint has nothing to do with ...
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2answers
107 views

Is there a surface in Euclidean space that admits elliptic geometry?

As I understand, on a pseudosphere, a surface of constant negative curvature, we can realize a part of the hyperbolic plane (but not the entire plane due to Hilbert's 1901 theorem) and use this for ...
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33 views

“Fundamental region” for non-discrete Moebius groups.

Suppose we are given a discrete, faithful representation $\rho$ of $F_2=\langle a,b|\rangle$, the free group on two generators, into $\mathbb{P}SL(2,\mathbb{R})$, so that the quotient is homeomorphic ...
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1answer
41 views

Can I solve for the fractional volume of a hyperboloid?

This looks like a homework problem because it is. I'm stuck at the portion where I solve for fractional volumes. Suppose you are a part of a team designing a water tank in the shape of a hyperboloid. ...
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64 views
+100

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
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1answer
34 views

problem with isometry that doesn't preserves rays, i just don't understand

I try to teach myself a bit of non-euclidean geometry And I am a bit stumped by the following remark in George E. Martins "The foundations of Geometry and the Non-Euclidean Plane" page 217. extra ...
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1answer
42 views

Find a point on the Poincare plane

Find a point $P$ on the line $_{-3}L_{\sqrt{7}}$ in the Poincare plane whose coordinate (ruler) is $2$. Let $P =(x,y)$. The line is on the Poincare plane, so it is a semicircle on the upper ...
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3answers
45 views

classifications of isometries of $\mathbb{H}^2$

Let $\mathbb{H}^2$ be the hyperbolic plane and let $\phi \mathbb{H}^2$ be the boundary at infinity of $\mathbb{H}^2$. Let the union $\mathbb{H}^2 \cup \phi \mathbb{H}^2$ be donoted by $\alpha ...
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1answer
35 views

boundary at infinity of $\mathbb{H}^2$

In hyperbolic geometry what does it mean when they say the boundary at infinity of $\mathbb{H}^2$? The only idea I came up with was a horizontal line to represent the horizon and to lines meeting at a ...
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1answer
27 views

Relation for hyperbolic pentagon.

I am trying to get a relation between the length of the sides and the angles of a hyperbolic pentagon. In literature I can find relations for pentagons which has at least three Right angle. So my ...
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15 views

progression along geodesics

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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2answers
118 views

List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.

I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation. One can find the ...
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0answers
12 views

small deviations along hyperbolic G-orbit quasi-geodesics

Let $A$ and $B$ be two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$. If $p$ is a point in the hyperbolic plane, we can consider the broken geodesic ray $\gamma_A$ described by the vertices ...
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70 views

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...
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1answer
42 views

How to construct hyperbolically equidistant points on a line?

In Stillwells' "Sources of Hyperbolic Geometry " page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line. I tried to reconstruct the figure but did ...
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1answer
33 views

Can an ordinary point be a fixed point?

Is it possible that a point on the unit circle which is an ordinary point (that is, a point which is not a limit point of any set of the form $\Gamma z$ for $|z| <1$) for a Fuchsian group $\Gamma$ ...
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0answers
85 views

What is the Area formed when a line is traced between two 3D curves?

This question is quite related to intersection of cylinders, Hyperbolic paraboloid and modelling. I am welding a trunnion to a pipe (both are hollow cylinders in different geometry). They intersect ...