Questions tagged [hyperbolic-geometry]

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.

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Hyprbolic geodesic attempt and maximum distance

Problem: Let $z,z',w,w'$ be points in $\mathbb{H}^2$. Let $w\in [z,z']$. Then $d(w,w')\leq max \{ d(w,z),d(w',z) \}$ Where $d= d_{\mathbb{H}^2}$ is defined in terms of the cayley map between poincare ...
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Properties of pseudospherical circles and geodesics

I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) and I have some questions about pseudospherical circles at page 10: the author says that Beltrami ...
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The Branch Schema of a Subgroup of the Modular Group

Let $\Gamma$ be a subgroup of the modular group $PSL(2, \mathbb{Z})$. What is the best and easiest way to grasp the notion of the Branch Schema of the subgroup $\Gamma$. Why do we have only four cases ...
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Why do isometries of Beltrami-Klein model correspond to projective transformations fixing the circle at infinity?

I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) by Nicola Arcozzi, and I am not quite understanding what is happening at page 9, in discussing ...
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44 views

The hyperbolic plane $\mathbb{H}^2$ can't be isometrically immersed in $\mathbb{R}^3$

It's easy to note that there is a local isometry between the hyperbolic plane $\mathbb{H}^2$ and the pseudosphere, since they have constant curvature equal to $-1$. Hilbert's Theorem.- There exists no ...
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1answer
55 views

The Norm of a Differential Form on $\Gamma\backslash\mathbb{H}$

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. Let $\mathbb{H}$ be the upper half-plane. Let $\phi$ be a cusp form of weight $2$ for $\Gamma$. Then $\omega=...
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46 views

What is the relationship between $S0(2)$ and $PSL(2,R)$?

The Holonomy of a hyperbolic surface S in terms of differential geometry is either $SO(2)$ or $O(2)$ depending on Orientability. And a hyperbolic structure as a special (X,G)-structure: $\pi_1(S)⊂PSL(...
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Showing that the sum of the angles of a hyperbolic triangle are less than $\pi$

Using the law of cosines for sides: $\cosh(a) = \cosh(b) \, \cosh(c) - \sinh(b) \, \sinh(c) \, \cos(\alpha) $ I have to show that $\alpha + \beta + \gamma < \pi$ Unfortunately I find no ansatz for ...
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hyperbolic quadrilateral angle

Consider a hyperbolic quadrilateral of $abcd$ in the hyperbolic plane $\mathbb{H}^2$ with the metric being the metric defined via the cayley map. Suppose $\angle b$, $\angle c$ ,$\angle d$ are all ...
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The Projective Special Linear Group and The Isometry Group of the Upper Half-Plane [closed]

Let $\mathbb{H}$ be the upper half-plane and $ISO(\mathbb{H})$ be the isometry group of $\mathbb{H}$. Let $PSL_2(\mathbb{R})$ be the projective special linear group. Is $PSL_2(\mathbb{R})=ISO(\mathbb{...
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What does it mean to compute the horizontal line segment between two points on a hyperbolic plane? [closed]

Compute the hyperbolic length of the following: the horizontal line segment between $(0,1)$ and $(1, 1)$. How am I supposed to do this using integrals?
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Properly Discontinuous Group Actions [closed]

Let $G$ be a group acting on a topological space $X$. We say the action of $G$ on $X$ is properly discontinuous (or $G$ acts properly discontinuously on $X$) if one of the following two equivalent ...
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How to convert from Euclidean to Hyperbolic Space using the Poincare Ball?

The Poincaré ball is $\mathcal{B}^d = \{x \in \mathbb{R}^d : \|x\|<1 \}$. Then the Poincaré model is given by the ball and the Riemannian metric tensor $$ g_x = \left(\frac{2}{1-\|x\|}\right)^2g^E$$...
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Busemann functions and inequalities

Let $D$ is the hyperbolic unit disk. Let $\alpha,\,\beta\in S^1$, where $S^1$ is the boundary of $D$. Let $w\in D$. I know that Busemann function for hyperbolic disk is $$B(w,\alpha)=\ln\frac{1-|w|^2}{...
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cusp shapes from vertex invariants

I am looking for an algorithm for computing the cusp shape from the vertex invariants of a complete triangulation of the toroidal cusp neighborhood of a knot complement. I have spent a fair amount of ...
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Analogous Euler line of a hyperbolic triangle

What curve passes through the circumcentre C, orthocentre O and centre of gravity G of a hyperbolic triangle? Would it be hyperbolically straight? Any ratio of hyperbolic distances among them? If the ...
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What is Representation of Surface Groups?

I had a question in my mind for a month ago. Mainly I am interested in Hyperbolic Geometry. I found a topic named "Representation Theory of Surface Groups". Let me tell about what is a "...
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Find a coordinate transformation of the hyperbolic plane for a non-standard representation

There are three common ways to represent the hyperbolic plane. One usually starts with the hyperboloid $x_1^2+x_2^2-x_3^2=-1$ embedded in a Euclidean space with negative signature $ds^2 = dx_1^2+dx_2^...
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Are there any “simple and intuitive” models of hyperbolic geometry?

I've read many times that hyperbolic geometry is geometry on a negatively curved surface, but when I try to research it online, I usually get things like the Poincaré disk or Beltrami-Klein disk, ...
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43 views

Transformation between different hyperbolic plane coordinate systems

On the hyperbolic plane, we can use 'polar' coordinates $(\rho,\tau)$ giving the metric $$ds^2 =d\rho^2 + \sinh^2(\rho) d\tau^2,$$ or, in the upper-half plane model, we can use the 'Cartesian' ...
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References on hyperbolic geometry and Teichmuller Theory

I am asking a soft question here. I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
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23 views

Normalised Liouville Measure on $\Gamma / PSL(2,\mathbb{R})$

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2, \mathbb{Z})$. What does Normalised Liouville Measure on $\Gamma / PSL(2,\mathbb{R})$ mean? Suggesting a reference is ...
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Geodesics in the upper half plane

There is a nice description of the geodesics on the upper sheet of the hyperboloid $H^2 \subset \mathbb{R}^3$ in terms of hyperbolic sine and cosine as $$ \lambda(t)= (\cosh(t))x + (\sinh(t))y $$ ...
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How to glue hyperbolic manifolds?

I want to understand the details of gluing for hyperbolic manifolds. By gluing I mean something along the lines of the statement that "a Riemann surface is glued together on the overlap of local ...
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64 views

The real part of $f(z)dz$ for a holomorphic complex function $f$ defined on the upper half-plane $\mathbb{H}$

Let $f$ be a holomorphic complex function defined on the upper half-plane $\mathbb{H}$, i.e., $f:\mathbb{H} \rightarrow \mathbb{C}$. Then, $f(z)dz$ can be written as follows $$f(z)dz=(u(x,y)+iv(x,y))(...
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72 views

$PSL(2,\mathbb{R})$, $PSO(2)$ and Hyperbolic Distance

Let $PSL(2,\mathbb{R})$ be the Projective Special Linear Group and $PSO(2)$ be the Projective Special Orthogonal Group. It is well-known that $PSL(2,\mathbb{R})/PSO(2)$ can be identified with the ...
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48 views

Classify the conjugacy classes in the group $PO(n,1)$.

Let $H^n=\{X=(x_1,x_2....x_n) \in\mathbb{R}^{n,1}: <X,X>=-1, x_1>0\}$ denote the upper hyperboloid in the hyperboloid model of the hyperbolic space. The components of $O(n,1)$ that preserves $...
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58 views

Harmonic $1-$ form on the upper half-plane $\mathbb{H}$

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. A function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire modular form for the subgroup $\Gamma$ ...
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39 views

Second order differential equation with hyperbolic function

Consider the second order differential equation $\frac{d^2f}{dt^2}+a\frac{df}{dt}+bf=0$, where $a,b\in\mathbb{R}$. For which values of $a,b$ do we have $f(t)=\sinh(At+B)$ being a solution of this DE? ...
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35 views

Some version of MVT(?)

Let $T \in SL(2,C)$ be a normalised Mobius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$. The above is an exercise from Outer Circles by Marden (ex. 1.1). I ...
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$PSL(2,\mathbb{R})$ and the set of oriented geodesics

How can $G/D$ be identified with the set of (oriented) geodesics?
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26 views

Hyperbolic spaces

Suppose X is a δ-hyperbolic metric space and α is a geodesic in X. Let P : X → α denote any nearest point projection map, i.e. for all x ∈ X, d(x, α) = d(x, P(x)). Show that the P coarsely L-Lipschitz ...
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Random walks on the Poincaré disk

Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \mathbb{C}, |...
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51 views

Doubt in the Proof on Dehn-Nielsen-Baer theorem

I'm reading the proof of Dehn-Nielsen-Baer theorem given in the book 'A primer on mapping class groups' by Margalit and Farb. I'm having trouble understanding the following line from the proof of ...
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Locally geodesic but not quasi geodesic

A path p is called a K-local geodesic if for all x and y in p, $d_{p}(x, y) \leq$ K implies that $d_{p}(x, y) = d(x, y)$. For all K, give an example of an infinite K-local geodesic in Z⊕Z which is ...
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Gaussian Integer and Ford Domain

I have been thinking a problem for a few days, but I can not find a solution, I hope you can help me. Let me please to present the problem and some definitions. Problem: Describe the isometric spheres ...
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When $z \to 0$, can we approximate the hyperbolic distance between $z$ and $0$ by the usual Euclidean distance?

Let $z \in \mathbb{C}$. Let $\rho$ represent the hyperbolic metric on the open unit disk and $|z|$ be the usual length of $z$, when $z \to 0$, can we say that $\rho(z , 0) \to |z|$? I just want to ...
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47 views

Is it possible to find isometric mapping of discrete points on two hyperbolic spaces with different curvature?

I understand that hyperbolic spaces with different curvatures are not isometric. However, I wonder if this is also true for finite, discrete points on two hyperbolic spaces with different curvatures. ...
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65 views

Reference request for 3-manifold

I am asking a soft question. I am planning to learn $3$-manifold using the book "Geometry and topology of three-manifolds" by William Thurston. I want to know how much of Riemannian geometry,...
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1answer
36 views

zero map of hermitian forms

$n \times n$ matrix $A$ with complex entries is called Hermitian if $A^{*}=A,$ where $A^{*}=\bar{A}^{T}$ $$ H(\Bbb{C}):=\left\{A \in M_{2}(\Bbb{C}) \mid A^{*}=A\right\} $$ $H(\Bbb{C})$ consists of 2 ...
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Find point for geodesic on two-sheeted hyperboloid

The two-sheeted hyperboloid $p$ is defined by $x^2 + y^2 - z^2 = -1$ We are given two points, $a = (x_a,y_a,z_a)$ and $b = (x_b,y_b,z_b)$. $a$, $b$ and $c$ defines the plane $q$. How can we choose a ...
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Reference Request in 3-manifolds, Teichmüller theory etc.

My interests are centered about hyperbolic geometry. I am keen to pursue my further study in mathematics in the following fields: The geometry of $3$-dimensional hyperbolic and anti-de Sitter ...
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32 views

Cusps of the modular forms

According to the definition of the cusps attached above the set of cusps is infinite and to be more precise it is $Q \cup \infty$ since $G$ is a subgroup of $SL(2,Z)$ so the identity matrix (that is a ...
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44 views

The Riemann Surface $G/H$

Let $G$ be a subgroup of $SL(2,Z)$ that is of finite index and $H$ be the upper half-plane. How is the quotient topological space $G/H$ defined (understood)?
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Highly recommended references on Modular Forms

I was reading this following attached below part of the asymptotic winding of the geodesic flow on modular surfaces and continuous fractions article, Y. Guivarc'h and Y. Le Jan. To be honest, I know ...
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1answer
42 views

Reference request for 3-manifolds [closed]

I am eager to study the following subjects. I am looking for which background I need to learn these. Also, I am looking for a learning roadmap of these subject. I want study " The geometry of 3-...
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1answer
56 views

Some simple questions regarding Modular Spaces, Modular Forms, Hyperbolic Geometry, Projective Special Linear and Modular Groups

Asymptotic Winding of the Geodesic Flow on Modular Surfaces and Continuous Fractions. Y. Guivarc'h and Y. Le Jan. I have been struggling with the first section (1. Framework and notations) for a long ...
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23 views

Transform point in the Poincaré disc to point in tile

Given a tiling of the hyperbolic plane with a finite set of tiles, how can I transform a point in the Poincaré disc model to the tile that occupies that point and get the corresponding point in that ...
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1answer
46 views

Length of a split complex number? [closed]

A complex number can be written as follows: $$ a+ib=re^{i\theta}, \text{ where } i^2=-1 $$ and where $r=\sqrt{a^2+b^2}$. A split complex number can be written as follows $$ a+jb=re^{j\theta}, \text{ ...
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57 views

Isometry of open set in upper half plane with hyperbolic metric

While reading Donaldson's book on Riemann surfaces, I came across the following proposition. Proposition Let $H$ be upper half plane, $p,\tilde{p} \in H$, and $f: N \rightarrow \tilde{N}$ be a ...

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