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

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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
40 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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33 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
123 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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23 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
97 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
30 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
63 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
89 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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48 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
95 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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135 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
279 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
118 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
47 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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29 views

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
107 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
45 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
41 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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63 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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55 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
111 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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92 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 ...
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1answer
49 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
68 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
267 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
125 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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40 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
46 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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117 views

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
38 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
45 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
53 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
39 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
30 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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16 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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160 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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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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135 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
54 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
37 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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93 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 ...
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5answers
436 views

Distance between points in hyperbolic disk models

I was puzzeling with the distance between points in hyperbolic geometry and found that the same formula is used for calculating the length in the Poincare disk model as for the Beltrami-Klein model ...
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81 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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2answers
728 views

The shape of Pringles potato chip

Why the shape of Pringles potato chip is hyperbolic paraboloid? I found several articles that say the shape is hyperbolic paraboloid, but cannot find out why it is so. Does anyone have reasonable ...
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1answer
50 views

What does the logarithm of a hyperbolic line look like?

At a fixed point $p$ of hyperbolic $n$-space $H$, there is the exponential map from flat $n$-space to $H$ taking straight lines through the origin of the flat space to hyperbolic lines through $p$ in ...
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126 views

Please explain the shortest path between two points in non-euclidean geometry. [closed]

Please explain it for those with inferior knowledge of mathematics (using easy to understand words): e.g., kids and adults with no knowledge of mathematics (calculus, algebra , etc.) , or rather from ...
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0answers
50 views

Internal angle formula of a generalized polygon as a function of side length and apothem

I am looking to compute the internal angle of a generalized regular polygon (spherical, euclidean, or hyperbolic) as a function of its apothem and side length. I know the equation for a euclidean ...
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1answer
51 views

Is the usual topology on the upper half plane same as that induced by Riemannian metric?

The upper half plane is a Riemannian manifold, with the Riemannian metric given by $(ds)^2 = (dx^2+dy^2)/y^2$ and thus has a metric topology induced by this metric. Is this topology same as the ...
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83 views

Distance from a point to a line in the hyperbolic plane

I have two questions: What is the distance from a point to a line in the hyperbolic plane? Fix a line $L$ in the hyperbolic plane. What does the set of points of distance $d$ from $L$ look like?