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

learn more… | top users | synonyms

5
votes
1answer
546 views

Geometric construction of hyperbolic trigonometric functions

If we have a circle we can geometrically construct the trigonometric functions as shown. The functions all derive from sin and cos. If we say that the circle is a conic section and imagine it on the ...
2
votes
1answer
120 views

Seifert manifolds and Fuchsian group

A Fuchsian group is a discrete subgroup of $PSL(2, \mathbb{R})$. Let $M$ be a Seifert manifold (maybe with boundary) and $t \in \pi_1(M)$ the class of a regular Seifert fiber. Hempel claims in his ...
2
votes
1answer
183 views

Questions about triangles, n-gons, and tessellations of the hyperbolic plane

Why there are infinitely many regular tessellations of the hyperbolic plane? Can there be a triangle made up of three straight lines in the hyperbolic plane? I know it's impossible since the angle ...
0
votes
0answers
56 views

Drawing graphics on a pseudosphere

I'm pretty sure this question is going to be very difficult to answer, so I will do my best to explain my problem, and maybe, just maybe someone will have a good answer. I have written a program ...
0
votes
1answer
75 views

What is complex conjugation in Hyperbolic Geometry

Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite ...
1
vote
1answer
108 views

justifying reflection across line in beltrami-klein model

Justify the following construction of the Klein reflection A' of A across m. Let Λ be an end of m and P be the pole of m. Join Λ to A and let this line cut y (which is the circle, my note) ...
2
votes
1answer
53 views

Simple closed geodesic around two hyperbolic cusps.

Consider a connected hyperbolic $2$-manifold $M$ with cusps. Consider a simple closed geodesic in $M$, which dissects $M$ into two components. Assume that one of the components contains exactly two ...
1
vote
1answer
78 views

projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
0
votes
1answer
44 views

Verifying the vertices of a Hyperbola

In this exercise it is required to verify that the points O & A(4,0) are the vertices of the Hyperbola H , as you can see in the marked part. Can someone help verify that ?
4
votes
1answer
61 views

Computing the volume of a hyperbolic knot

Could anyone show me or refer me to a link where the volume of a hyperbolic knot, say, the figure-8 knot, is computed (well, in fact estimated) explicitly and not only having the procedures outlined?
2
votes
1answer
84 views

Quotient under a Fuchsian Group

Let $D$ be the unit disk in the complex plane, and let $H$ be a fuchsian group generated by one fixed point free element, say $a$. What it the quotient $D/H$? Attempt: The quotient is biholomorphic ...
2
votes
1answer
75 views

A Fuchsian Group?

Let $p_k := e^{\pi/2 i k}$, $k \in \{0, 1,2,3\}$. Let $b_k$ the geodesic of the hyperbolic disk connecting $p_k$ and $p_{k+1(\text{mod}4)}$. For instance, $p_0$ and $p_1$ are connected by the lower ...
1
vote
2answers
143 views

Hyperbolic Geometry - reference request

I need some information about Hyperbolic Geometry. For example, Spherical Geometry is a subsection of Hyperbolic Geometry or no? Can you suggest to me a book or some other reference to help me ...
3
votes
0answers
77 views

Question regarding the projective models of the anti-de-Sitter spaces and good online references for learning them from the scratch? (Specifics below)

As my title says above, I am trying to find answers to and also good online reference where I can find complete description of projective models of hyperbolic space, de-Sitter space and anti-de-Sitter ...
2
votes
1answer
113 views

Isometries of a hyperbolic quadratic form

I am reading an article that says "The group of isometries (of a hyperbolic space) of a hyperbolic quadratic form in two variables is isomorphic to the semi-direct product $\mathbb{R} \rtimes ...
2
votes
1answer
127 views

Isometry fixing two points of a geodesic line

Let $H$ be a hyperbolic space, and let $\Gamma \subset H$ be a geodesic line, i.e., the image of an isometry from $\mathbb{R}$ to $H$. If $f$ is an isometry of $H$ that fixes two distinct points of ...
2
votes
1answer
191 views

Arc length parameter s

Consider the metric $$ds^2 = \frac{dx^2+dy^2}{y^2}.$$ Assume $R>0, a\in\mathbb{R}$. Consider the curve $$\gamma(\theta)=(a+R\sin\theta,R\cos\theta)$$ for $-\frac{\pi}{2}\leq\theta\leq ...
2
votes
2answers
155 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
0
votes
1answer
95 views

Möbius transformation that preserve distance for two pair of given points in $\mathbb{H}$.

I need to prove that for a given two pair of points $(z_1,z_2)$ and $(w_1,w_2)$ in $\mathbb{H}$ (Poincaré's upper half plane), where $d_{\mathbb{H}}(z_1,z_2)=d_{\mathbb{H}}(w_1,w_2)$, there is an ...
4
votes
1answer
222 views

Study of the Laplacian on the Hyperbolic plane

What's a good reference for the simplest case? I'm interested in the spectral theory of the Laplace-Beltrami operator on the upper half plane (domain, self-adjoint extension, etc.). I only need this ...
0
votes
2answers
224 views

circle reflections in hyperbolic geometry

Determine the equation of the circle reflection of the circle $x^2 + y^2 = 1$ if the circle of reflection is $x^2 + y^2 + 2x = 0$. I'm learning about circle inversion but I still don't get what this ...
2
votes
1answer
44 views

Need help with finding length of sides and angles of a triangle in upper half plane model

The given points are $i, 3i, 1 + 2i$ I know that the distance for points on a vertical line can be found by using the formula $$\ln\left|\frac{y_2}{y_1}\right|$$ So the distance between points $ i$ ...
1
vote
1answer
186 views

Finding an angle of a triangle in the upper half plane model given three points

I've been given three points in the upper half plane $(i, 3i, 1 + 2i$), and one of the homework questions that asked is to find the angles of the given triangle. A previous problem asks to find the ...
1
vote
1answer
96 views

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides?

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides? I have been trying to visually see if that is possible or not.
1
vote
1answer
49 views

Quick Hypberbolic Geometry question concerning Saccheri Quadrilaterals

Can a Saccheri Quadrilateral have 3 congruent sides? I know the summit is less then the base, but could it happen that the base is the same length as the two vertical sides?
1
vote
1answer
86 views

hyperbolic quadrilateral angles

On the hyperbolic plane, if I have a quadrilateral that has all congruent interior angles $\alpha$, how do I figure out what $\alpha$ is? I know in Euclidean geometry one could just use ...
1
vote
1answer
106 views

geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
1
vote
1answer
105 views

How does my Beltrami-Klein model look?

Did I sketch the picture right based off of the specific instructions given in the problem?
1
vote
1answer
161 views

convex polygons in hyperbolic geometry

Does $\exists$ on the hyperbolic plane, a convex quadrilateral $Q$ and a convex pentagon $P$ with the same angle sum? I found this question to be rather interesting.
1
vote
2answers
31 views

Hyperbolic quadrilaterals with two adjacent right angles

For convenience we'll work in the hyperbolic upper half plane $H$. We are given a hyperbolic quadrilateral $Q$ with vertices $a,b,c,d$ and geodesic segment edges $[ a,b ]$ $[ b,c ]$ $[ c,d ]$ $[ d,a ...
1
vote
0answers
212 views

Calculating hyperbolic distance between two points

I was looking for a formula to calculate the hyperbolic distance between two planes and came across this formula $$\ln\left(\csc b-\cot b\over \csc a - \cot a\right)$$ where $a$ and $b$ are the ...
1
vote
1answer
60 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
3
votes
2answers
131 views

Showing the function $f(x,y)$ is one by one

Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
1
vote
1answer
83 views

Projecting external points to a circle: Distance order preserving?

Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation: I compute the point of intersection of the i) circle and the ii) line joining each ...
7
votes
2answers
228 views

Simple non-closed geodesic.

In torus there exists simple non-closed geodesic. One example is take the irrational slope curves in $\mathbb{R}^2$ and project it down to torus. Is this thing can happen in closed hyperbolic surface ...
3
votes
1answer
87 views

Any hyperbolic $n$-simplex is contained in an ideal simplex

Recall that an $n$-simplex in $\overline{\mathbb{H}^n}$ (the closure of $n$-dim hyperbolic space) with vertices $v_0,...,v_n\in \overline{\mathbb{H}^n}$ is the closed subset of $\mathbb{H}^n$ bounded ...
0
votes
1answer
224 views

Hyperbolic geometry

Post Number: 45 Posted on Friday, 22 March, 2013 - 04:48 pm: I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this? Show that there ...
3
votes
2answers
145 views

topic for presenting in hyperbolic geometry

For my course work, i have to give a presentation of 20-30 min presentation in hyperbolic geometry. Can any one suggest some topic(or any interesting theorem) in this area.I want to present some thing ...
5
votes
1answer
90 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
1
vote
1answer
88 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
4
votes
1answer
83 views

Reference Request: Regge Symmetry “Angle-Edge” Duality

A tetrahedron in hyperbolic 3-space can be defined (up to isometry) by the measures of its dihedral angles, $(a, b, c, a^\prime, b^\prime, c^\prime)$, with $a$, $b$, $c$ along edges that meet at a ...
-1
votes
1answer
140 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...
1
vote
1answer
50 views

Questions on two elements in a Fuchsian group which have at least one common fixed point

This is a homework question I am unable to solve. Let $A,B \in PSL(2,R)=Aut(H)$. Assume none of them are elliptic and they have : Case 1) one common fixed point at the boundary of $H$ (i.e. they ...
3
votes
1answer
188 views

Set of points equidistant from two points in hyperbolic space.

Given two points $p,q$ in the hyperbolic plane, show that the set of points equidistant from $p$ and $q$ is a hyperbolic line. I am unsure how to proceed with this question. Would it be easier to use ...
3
votes
2answers
205 views

Reflection in a hyperbolic line formula

Let $H$ denote the upper half-plane model of hyperbolic space. If $L$ is the hyperbolic line in $H$ given by a Euclidean semicircle with centre $a\in \mathbb{R}$ and radius $r >0$, show that ...
3
votes
2answers
190 views

Measure on a quotient

Can anyone explain me the following : let $M$ be a hyperbolic manifold and $\Gamma = \Pi_1(M) \subset Iso(\mathbb{H}^n) $. How does the Haar measure on $Iso(\mathbb{H}^n) $ induces a measure on ...
5
votes
1answer
311 views

Möbius Transformations are Orientation Preserving?

This question is truly stupid, but is driving me crazy. I just need an outside viewpoint to sort out what's going on. In my textbook: "Show that every linear fractional (LF) transformation of ...
3
votes
1answer
253 views

Rigid body motion on the Poincare disc model of the hyperbolic plane

I'd like to implement an interactive simulation of an actor controlled by the user moving around in the Poincaré disc model of the hyperbolic plane. I need to know how to perform translation and ...
0
votes
0answers
28 views

Enlightening explanation of a theorem of Zimmert's

I'd like to know wether anyone has ever read an enlightening explanation (e.g. with geometric argument) of the following paper: Zimmert, R. Zur $SL_2$ der ganzen Zahlen eines imaginär-quadratischen ...
1
vote
1answer
50 views

Regular triangulation of compact oriented hyperbolic space

Is there a good way of explicitly constructing a regular triangulation of a compact orientable hyperbolic 2-manifold, ideally with any desired vertex degree $\ge 7$? I only need the topology, not any ...