Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.
4
votes
1answer
155 views
Characterization of linearity in terms of metric
At least in Euclidean geometry and the upper half plane model of hyperbolic geometry, the statements '$y$ lies on the line segment determined by $x$ and $z$ ' and '$d(x,y)+d(y,z)=d(x,z) $' are ...
2
votes
1answer
38 views
Right angles in hyperbolic pool
(This uses a bit of physics)
So I learned today the following fact from physics: Imagine you have two pool balls of the same mass. You hit the first one, and it collides into the second. Then their ...
2
votes
1answer
86 views
A Kleinian group has the same limit set as its normal subgroups'
It should be well known that a Kleinian group and all its normal (non-elementary) subgroups have the same limit set. Do you know any book/article where I could find the proof?
Thank you.
2
votes
1answer
68 views
Simple understanding of convex co-compactness
I was looking for the definition of the term "convex co-compact" in simple cases. But most references I find are looking into a little bit sophisticated notions such as mapping class group, Schottky ...
1
vote
1answer
58 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 ...
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 ...
1
vote
1answer
32 views
L is a family of hyp-lines passing through a pt. Not sure how this implies rr'=(c−Re(p))c'
Lemma 1. Let $p \in \mathbb{H}$, and assume $l$ is a family of hyp-lines passing through $p$ such that $l$ is of the form $l = \{c +re^{i\theta} | 0 < θ < π\}$. For simplicity, assume the ...
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?
0
votes
1answer
33 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.
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.
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
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
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 ...
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
...
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 ...
3
votes
0answers
88 views
How do we define a complete metric on a Riemann surface with punctures?
This question is related to another question.
If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?
I know that in this ...
3
votes
0answers
62 views
Construct tiling group from hyperbolic polygon
Given a hyperbolic $4n$-gon $P$ in the Poincaré disk, how can we construct explicitly the subgroup $G < \mathrm{Aut}{\left(\mathbb{D}\right)}$ which gives a tiling of $\mathbb D$ with fundamental ...
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 ...
2
votes
0answers
145 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 ...
2
votes
0answers
75 views
Distance realizing geodesic in a hyperbolic surface
Suppose $S$ is a hyperbolic surface with geodesic boundary and $P$ is a hyperbolic pair of pants with $a$, $b$, $c$ geodesic boundary. Let $\gamma$ be the unique geodesic realizing the distance ...
2
votes
0answers
144 views
Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?
In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces?
In ...
2
votes
0answers
35 views
ruling out non Pseudo-anosov automorphisms
We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
2
votes
0answers
146 views
The fundamental group of the mapping torus is doubly degenerate
Consider an hyperbolic compact surface $S$ (hence with genus $>1$) and a Pseudo-Anosov diffeomorphism $\varphi\colon S\to S$. We call "mapping torus" the 3-manifold ...
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 ...
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, ...
1
vote
0answers
139 views
Aristotle's Axiom in Hyperbolic Geometry
I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs ...
1
vote
0answers
121 views
Lines in coordinate system of Hyperbolic Plane
An orthogonal coordinate system of the hyperbolic plain can be set up by fixing an orgio $O$, an $x$-axis, a $y$-axis (intersecting each other at $O$ in angle $90^\circ$), and, from any point $P$ ...
1
vote
0answers
144 views
Is there non-discrete group isomorphic to the fundamental group, what about the quotient?
It is known that (uniformization theorem) any Riemann surface can be written as the quotient of its universal cover by a discrete group (of Möbius transformations). This group is isomorphic to the ...
1
vote
0answers
125 views
the hyperbolic plane is complete
I'd like to prove that the hyperbolic plane is complete in a short, nice way.
Here's my proof, but I'm not convinced of its correctness yet.
Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
1
vote
0answers
35 views
limit set of Kleinian grouos with closed manifolds as quotient
I'm trying to convince myself that if $M\cong\mathbb{H}^3/G$ is a closed hyperbolic 3-manifold then the limit set $\Lambda(G)$ equals the whole Riemann sphere $S_\infty^2$. My idea of the proof goes ...
1
vote
0answers
50 views
Relations between Kleinian groups and quotient manifolds
In some specific situation there are some nice relations between a Kleinian group and its quotient manifold. For example, if $G$ is a once-punctured-torus group (i.e. a free subgroup of ...
1
vote
0answers
63 views
For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset
Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve.
Let $P$ be the identity element of $E$.
Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
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 ...
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 ...
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 ...
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 ...
0
votes
0answers
81 views
The lambda invariant
Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane.
...
