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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Show the indefinite unitary group acts transitively on the hyperbloid manifold

Consider an indefinite Hermitian form $\langle \cdot ,\cdot \rangle$ on $\mathbb{C}^{n+1}$ such that $$\langle v ,w \rangle = \sum^n_{i=1} v_i \bar{w}_i - v_{n+1} \bar{w}_{n+1}. $$ We let $U(n,1)$ ...
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How (if!) can we connect two 2D hyperbolic manifolds by a "wormhole" structure? [closed]

Imagine two 2D hyperbolic manifolds. I connect them by a manifold like one sees pictured in images of a wormhole in 2D. Can we glue them together so the geodesics always diverge? I mean, we can view ...
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For the reflection $\phi$ in the sphere $S(a,r)$ it holds that $\phi(B^n) = B^n$ if and only if $\phi(a*) = 0$

This is a question about Theorem 3.4.2 in "Geometry of discrete groups" from Beardon. Let $\phi$ be the reflection in a sphere $S(a,r)$ for $a \in \hat{\mathbb{R}^n}$, $r \in \mathbb{R}$. ...
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Ultrametric spaces are $0$-hyperbolic

Let $(X, d)$ be an ultrametric space. In particular, X satisfies the strong triangle inequality: for any $x, y, z \in X$, we have $$d(x,y) \leq \max\{d(x,z), d(y,z)\}.$$ I want to show that $X$ ...
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The figure eight knot complement in $S^3$.

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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The Definition of Hyperbolic Triangle Group [closed]

Outer Circles, An Introduction to Hyperbolic $3-$Manifolds. Albert Marden. Page $86$. A Fuchsian group $\Gamma$ is called a (hyperbolic) triangle group of signature $(p,q,r)$, $2 \leq p,q,r \leq \...
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The hyperbolic angle in the Beltrami-Klein model is not necessarly the euclidean angle

I am trying to prove this result : The hyperbolic angle between two vecteors $u,v\in T_z\mathbb K$ in the Beltrami-Klein model $\mathbb K$ is not necessarly the euclidean angle between this two ...
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Simple formulas for geodesics of the Beltrami-Klein model

Exercice : Show that the geodesics of the Beltrami-Klein model are the euclidean lines ? Let $\gamma_{p,T}(t)=\cosh(t)p+\sinh(t)T$ be a geodesic of the hyperbolic plane such that $\gamma_{p,T}(0)=p=(...
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How to define a (linear, invertible) mapping from a square to a triangle

This problem has come up when analyzing one type of HSV color selector: Note: h,s,v are in 0...1 range. For the given hue (i.e. red) on the three vertices of the triangle we have: W: white, when s=0,...
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Isolated edges in hyperbolic Delaunay triangulation

I have played with the C++ library CGAL to do some hyperbolic Delaunay triangulations. Sometimes (often) the triangulation has some isolated edges, as in this example: Is it theoretically normal to ...
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Calculating the area of the hyperbolic triangle with vertices at $e^{i\pi/3}$, $i\sqrt2$, $i$ in the Upper Half Plane model

Sketch the geodesic triangle in H with vertices at $e^{i\pi/3}$, $i\sqrt2$, and $i$, and calculate its area. I'm not sure how to go about this question, any help would be great, thanks. I know I need ...
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How does the following definition of hyperbolic arc length apply to distance problems? (Differential Geometry)

I have some doubts about the application of the following. Let be the regular surface $\mathbb{H} =$ {$(x,y,z)$ $\in \mathbb{R^3} | z=0, y >0$}. For each point $p =(x,y,0) \in \mathbb{H} $, define ...
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Relationship between Axiomatic geometry and Hyperbolic geometry

I have done quite a bit of hyperbolic geometry and euclidean geometry, but one thing remained obscure to me throughout: the connection between axioms and the metrical definition of geometry. As I ...
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Need help with hyperbolic geometry

The problem is In △ABC, it is given that $\operatorname{m}(\measuredangle ACB) = 90^\circ$ and $AC = BC$. Let $M$ be the midpoint of $AB$. Prove that $CM < AM$. I know that you can prove this by ...
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As a function of $a$, how many points are there in hyperboloid $x^2 − y^2 − z^2 = 2$ where the tangent plane is parallel to plane $z-ax=3$?

PROBLEM As a function of $a$, how many points are there in hyperboloid $x^2 − y^2 − z^2 = 2$ where the tangent plane is parallel to plane $z-ax=3$ ? MY APPROACH I started by finding the normal vector ...
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$d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$.

I'd like some assistance in demonstrating $d_{\mathbb{H}}(p,q)=|\log(pq;rs)|$ where $\mathbb{H}$ is a Poincare Upper Half-Plane, and $(pq;rs)$ is a cross ratio of $p,q,r,s$. ($p,q$ are on a geodesic ...
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Two horocycles with the same center are equidistant.

We say two circles $C_1, C_2$ are equidistant if $d(p,C_2)=K$ for any $p \in C_1$ and $d(q,C_1)=K$ for any $q \in C_2$, where $K$ is some constant. I was told that if I use the Upper-Half plane model (...
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Good books for Poincaré and Klein trigonometry

I am looking for good reference to study the trigonometry of Poincaré and Klein disks, without the approach of Ungar is his books. Any help is really appreciated !
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Interpretation of the equation of a horocycle in the hyperboloid model.

Let $v$ be non-zero with $B(v,v)=0$. I would like to show that an equation $B(u,v)=1$ defines a horocycle in the hyperboloid model. A horocycle in the Poincare disk model has a touching point $v$ on ...
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How to get a conformal factor between the Poincare Disk's metric and the Euclidean Metric?

Conformal Factor. As you see in the linked post, the conformal factor is $\frac{2}{1-\left\|x \right\|^2}$. But I'm more interested in how the outcome turns out. According to what I've heard, this may ...
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Does every triangle in the hyperbolic plane have an incircle?

It is known that NOT all triangles on the hyperbolic plane have a circle that contains the triangle and passes thru all its 3 vertices. IOW, the circumcircle is not a universal property of triangles. ...
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Express a point $v$ in terms of $v_1$ and $v_2$ and the hyperbolic distance.

In the hyperbolic plane, we have this following result : On the geodesic boundary $\left(v_{1}, v_{2}\right)$, for example, we have, for $v \in\left(v_{1}, v_{2}\right)$, $$ v=\frac{\sinh \left(c-d_{\...
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Hyperbolic barycentric coordinates : Show that $\sum_{i=1}^3 \frac{\sinh \varepsilon_{i}}{\sinh \left(\varepsilon_{i}+\mu_{i}\right)}=1.$

Let $\mathbb H^2$ be the hyperbolic space and consider a hyperbolic triangle $(v_1,v_2,v_3)$ and a point $v \in \mathbb H^2$ as showed in the figure : Let $\mu_i$ is the hyperbolic distance from $v$ ...
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Diffeomorphism from Minkowski hyperbolic plane to Poincaré Disk

$\newcommand{\cyclic}[1]{\langle #1\rangle}$ This is a seemingly straightforward question from Galot, Hulin, Lafontaine: Riemannian Geometry, Third Edition. We have a model of hyperbolic space ...
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Haar measure on $PSL_2(\mathbb{R})$ via Iwasawa decomposition

I am missing something basic about the relation between the Haar measure on the group $G = KAN$ and the haar measures on the subgroups $K$, $A$, and $N$. Specifically, let $G=PSL_2(\mathbb{R})$ then ...
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How to find a formula of a reflection in hyperbolic geometry.

I'm looking for a hyperbolic geometry formula for a reflection. However, I only know how to find a reflection point and not a formula, so I'd appreciate some assistance. Let $B$ be the quadratic form $...
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Prove that there's an element in $O^+(1,2)$ that sends one given line to another given line.

This is my first encounter with isometries of hyperbolic geometry. As a result, there may be some issues with my comprehension of the concept. But first, let me explain how, with my limited ...
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In Thurston's notes on hyperbolic geometry, why does a geodesic always belong to a 2D subspace?

On page 12 of [1] (in the subsection titled Trigonometry), Thurston derives the equation of a geodesic in the hyperboloid model of hyperbolic geometry. The hyperboloid in the hyperboloid model is ...
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Find the real and imaginary parts of $\sinh(e^i)$ (verify solution)

Find the real and imaginary parts of $\sinh(e^i)$ my attempt: $\sinh(e^i) = \frac{e^{e^i}-e^{-e^i}}{2}$ $= \frac{e^{\cos(1)+i\sin(1)}-e^{-\cos(1)-i\sin(1)}}{2}$ $= \frac{e^{\cos(1)}e^{i\sin(1)}-e^{-\...
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Do Carmo Chapter 8 Exercise 2

The statement is: Show that if $M^k$ is a closed, totally geodesic submanifold of $H^n$ (hyperbolic $n$ space), then $M^k$ is isometric to $H^k$. My friend and I thought to use the "Cartan ...
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Categoricity of Hyperbolic Geometry

I have been reading George E Martin's classic text on Foundations of Geometry. In this book, the authors states that "axioms for hyperbolic geometry are not categorical" rather they are &...
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Simple formulas for geodesics of the Poincaré disk?

Let $D$ be the Poincaré disk endowed with the metric $g=4\dfrac{dx^2+dy^2}{(1-x^2-y^2)^2}$. I want to find a sample equation for the geodesics between two points $p$ and $q$ in the disk. We know that ...
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Triangulations of the Hyperbolic Plane

I'm studying triangulations of the hyperbolic plane and have come across the following theorem: If we are given a triangle $\Delta_0$ with angles $\pi$/l,$\pi$/m,$\pi$/n, where the integers l, m, n ...
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Recommended books for Beltrami-Klein Disk : [duplicate]

I am looking for good books to study the Beltrami-Klein Disk in Hyperbolic geometry because most of books that I have discuss just the Poicaré half plane, the Poincaré disk. I want that the book ...
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Primer on Mapping Class Groups Chapter 1: Arbitrarily short loops around Punctures

Dear fellow Mathematicians, This is the first question I ask in this forum, so please excuse any formal mistakes, which I am, of course, trying to avoid. I am currently briefly revisiting hyperbolic ...
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Warp product structure on the complex hyperbolic space

It is known that the real hyperbolic space $\mathbb{H}^n$ has a warped product structure $g = dr^2 + \sinh^2 r\; ds^2_{n - 1} $, where $r \in (0, \infty)$. My question is, does the complex hyperbolic ...
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Can we do regular geodesic tiling on a quadric surface, such as hyperboloid and paraboloid?

Can we do regular tiling on a quadric surface, such as hyperboloid and paraboloid? Assume that the straight lines are the geodesics. Can you show the pictures of the tiling?
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barycentre in hyperbolic geometry :

In euclidean geometry we know the formulas for midpoint and barycentre of a finite set of points, so can we find similar formulas in hyperbolic geometry ? In the Klein disk, Ungar cited in his book &...
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On the nature of mosaic specified by Schlafli symbol $\{p,q\}$

I was reading a paper on hyperbolic pascal triangle and the author stated that for Schlafli symbol $\{p,q\}$, if $(p-2)\;(q-2)=4$, it determines the Euclidean mosaic. For $(p-2)\;(q-2) <4$ a sphere ...
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Formula for action of $\operatorname{SL}(2,\mathbb{C})$ on hyperbolic 3-space [duplicate]

It's pretty standard in 3-manifold topology and hyperbolic geometry that $\operatorname{PSL}(2,\mathbb{C})$ is the orientation-preserving isometry group of hyperbolic 3-space $\mathbb H^3$. I haven't ...
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Tesselation of hyperbolic plane are hyperbolic

Let’s take a tesselation $T$ of the hyperbolic plane (not necessarily regular), my intuition tells me that clearly $T$ should be hyperbolic itself (in the sense of Gromov or using $\delta$-slim ...
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Geodesic Polar Coordinates and Area of a Circle in Hyperbolic Space

I am trying to calculate the area of a circle of radius $R$ in the hyperbolic plane. I know that the answer is supposed to be $4\pi(\sinh(\frac{R}{2}))^2$. However, I am not getting this. To calculate ...
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3 votes
1 answer
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Isometries of n-dimensional hyperbolic space

I'm aware that the isometries of hyperbolic 2 dimensional and 3 dimensional space are given by elements of $SL(2,\mathbb{R})$ and $SL(2,\mathbb{C})$ respectively and are then categorised according to ...
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Eulidean diameter of hyperbolic disk can be controled by distance from centre of the disk to the boundary

Let $\Omega \subset C$ be a hyperbolic domain, $a \in \Omega$, $B(a,d)$ is a hyperbolic disk centered in $a$ with radius $d$. To show that: there exists a constant $C(d)$ which depends on $d$ only, ...
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Finding omega points on Poincare’s disk model of hyperbolic geometry

I was trying to prove that "a hyperbolic circle in $\mathbb{D}^2$ is a Euclidean circle in $\mathbb{D}^2$ and vice-versa" ($\mathbb{D}^2$ is the Poincare's disk) then I think that I need to ...
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Tiling of Klein quartic with 56 equilateral triangles

I am interested in the regular hyperbolic 14-gon associated with the Klein quartic (Klein's famous "Hauptfigur"). This 14-gon, or more precisely the genus-3 surface obtained by identifying ...
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What is the locus of all shape parameters of ideal tetrahedra which share the same volume?

I'm taking a class in hyperbolic knot theory out of Jessica Purcell's book, and I was curious about some volumes and classifications of ideal tetrahedra in $\mathbb{H}^3$, with the upper half $n$-...
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Motivation for hyperbolic metric on upper half plane

How to motivate the definition of distance in the upper half plane model of hyperbolic geometry? The book I am using just throws out complicated looking formulas involving ln, with no motivation. I ...
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Deviation lines in Poincaré - disk model

The question has been asked before I guess; two geodesics are given in Poincaré disk model say $K=-1$ cut by a transversal $(T_1T_2)$. Please help find common yellow area and perimeter enclosed ...
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Why Euclidean space isn't (Gromov) Hyperbolic?

I'm trying to understand why the hyperbolic plane is Gromov Hyperbolic while the Euclidean plane isn't. I guess that there is connection to the fact that every similar triangles in $\mathbb{H}^2$ are ...
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