Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.
2
votes
1answer
62 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
43 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
24 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 ...
0
votes
1answer
26 views
Help with hyperbolic geometry problem
construct two lines through the point $(3,1)$ that are parallel to the line $x=7$
construct two lines through the point $(3,1)$ that are parallel to the line $x^2 + y^2=36$
4
votes
1answer
31 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
1answer
52 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
31 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
61 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
32 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.
0
votes
1answer
25 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
27 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
39 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
43 views
How does my Beltrami-Klein model look?
http://imageshack.us/photo/my-images/109/hyperbolicquestion.png/
Did I sketch the picture right based off of the specific instructions given in the problem?
0
votes
1answer
32 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
24 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
96 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
0answers
40 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 ...
4
votes
2answers
67 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
32 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 ...
0
votes
1answer
17 views
conformal mapping
Prove that application ζ^-1:B^2---> H^2 is conform.
ζ^-1(x1,x2,x3)=(x1/1+x3 , x2/1+x3)
B^2 is unit disc and H^2 is hyperbolic 2-space and conformal mapping preserve angles.
3
votes
0answers
53 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
35 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
0answers
23 views
hyperbolic circle, horocycle and hypercycle [duplicate]
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 exists a tangent hyperbolic straight line at every point on a ...
0
votes
0answers
124 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
1answer
70 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
53 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
57 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 ...
2
votes
0answers
55 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
92 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
30 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
106 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
89 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
45 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 ...
3
votes
1answer
59 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
117 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 Poincare disc model of the hyperbolic plane.
I need to know how to perform translation and ...
0
votes
0answers
22 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
34 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 ...
3
votes
1answer
87 views
Understand the Hyperbolic space
I've been trying to find the expression for the metric of the hyperbolic n-space, $\mathbb H^n$.
For $n=2$ I've found (e.g. here) that $$ds^2=\frac{dx^2+dy^2}{y^2}.$$
But for $n>2$ I can't seem to ...
3
votes
0answers
87 views
Axis of the product of two loxodromic isometries
Suppose that $X$ and $Y$ are two loxodromic isometries of the hyperbolic space and that the product $XY$ is also a loxodromic element.
We consider the axes of these three elements. I'd like to know if ...
5
votes
0answers
42 views
Hyperbolic diameter of Amsler's surface
I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
3
votes
2answers
241 views
What is the proof that rectangles do not exist in hyperbolic geometry?
I am in need of help figuring this out--
If the only straight lines in hyperbolic geometry are those that pass through the center, then isn't there a right angle? (horizontal and vertical) Which ...
0
votes
0answers
40 views
Fréchet mean of the hyperbolic shape space
The Fréchet mean of a general subspace is defined as
$$F(x)=\int_Mdist(x,y)^2d\mu(y),$$
where $\mu$ is the probability measure on a general metric space $(M,dist)$.
I understand that the Fréchet mean ...
3
votes
2answers
230 views
Construction of equilateral triangle in Poincare disc model
Points A and B are given in Poincare disc model. Construct equilateral triangle ABC.
Any kind of help is welcome.
0
votes
0answers
55 views
Sum of angles in a hyperbolic triangle with one ideal angle
I want to calculate the sum of the angles of the triangle formed in the hyperbolic plane from the points $(-1,1), (0,1)$, and $(1,1)$. This forms an angle at the origin which has an infinite slope for ...
2
votes
1answer
59 views
Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations
Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a
Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$
Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on
...
1
vote
1answer
56 views
Locally cyclic subgroups of a hyperbolic group
How can we show that locally cyclic subgroups (ie. groups whose finitely generated proper subgroups are cyclic) of a hyperbolic group are cyclic?
3
votes
0answers
101 views
Curvature of Hyperbolic Space
I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out?
I've been given the metric
...
1
vote
0answers
34 views
Is $M_g$ a subvariety of $M_{h}$ for some $h>g$
Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$?
If the answer is not known, ...
2
votes
1answer
32 views
Projection on geodesic lines in $\mathbb{H}^n$
Good morning everyone,
I was wondering wether or not is the projection on a geodesic line in $\mathbb{H}^n$ $1$-lipschitz for the hyperbolic distance.
I asked myself this question because i ran ...
3
votes
0answers
54 views
reference request: “p-adic” presentation of surfaces
On several occasions I heart about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
