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

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Finding points on hyperbola [on hold]

Can someone please answer me if I can find points dotted on hyperbola and parabola in the attached images using the center point and anything that I need to use like (radius and angle in circle)? I ...
2
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1answer
26 views

Measuring distance on the Poincare disk

I've seen several different ways to measure distance on the Poincare disk i.e Riemann metric/manifold (which I don't understand). However the method we're taught is using $\tanh^{-1}$ and complex ...
2
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0answers
21 views

Parallel postulate in hyperbolic geometry question [on hold]

Let $L =\{y=0\}\cap\mathcal{H}^{2}$, and let $P=(3,2,2)$. Show that the parallel postulate fails in $\mathcal{H}^{2}$ by giving two lines $L',L'' \in \mathcal{H}^{2}$ with $P\in L',L''$ and $L\cap ...
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0answers
89 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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2answers
53 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
49 views

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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0answers
41 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 $0<r<1$ ...
3
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1answer
222 views

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 ...
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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
22 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 ...
4
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2answers
142 views

Structure of $x^2 + xy + y^2 = z^2$ integer quadratic form

The pythagorean triples $x^2 + y^2 = z^2$ can be solved in integers using rational parameterization of solutions to $x^2 + y^2 = 1$. It goes through $(1,0)$, then consider the line $y = -k (x - 1)$ ...
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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
103 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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1answer
49 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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1answer
24 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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0answers
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
24 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: ...
2
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1answer
84 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
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 ...
2
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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
36 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
176 views

What is the relationship between hyperbolic geometry and Einstein's special relativity?

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic ...
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1answer
24 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
53 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
72 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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0answers
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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1answer
143 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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0answers
126 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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0answers
48 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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2answers
38 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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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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0answers
19 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 ...
2
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1answer
37 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
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 ...
2
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2answers
57 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 $p=(a,0)$, $a>0$, ...
5
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2answers
89 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 ...
2
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1answer
61 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: ...
2
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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 ...
13
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2answers
211 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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0answers
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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3answers
46 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
42 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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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
43 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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1answer
36 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 ...
3
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1answer
262 views

Thrice-punctured sphere

This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45: the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric. I ...
4
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2answers
121 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 ...