Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.
78
votes
4answers
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Hyperbolic critters studying Euclidean geometry
You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise.
...
18
votes
7answers
1k views
What are the interesting applications of hyperbolic geometry?
I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...
16
votes
1answer
394 views
How to create mazes on the hyperbolic plane?
I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the ...
10
votes
3answers
448 views
What is the connection of the sequence 3, 4, 5/3, 2/3, 1 with deep topics?
Quote from Don Zagier (Mathematicians: An Outer View of the Inner World):
" I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, ...
10
votes
1answer
242 views
Teichmüller spaces via representations
I don't have much expertise in this area but I am confused by a remark I overheard regarding Teichmüller spaces.
I was always under the impression that for a surface $S$ (say genus $\geq 2$) ...
10
votes
2answers
171 views
Embedding the Infinite Binary Tree in Regular Tilings
Consider the regular tiling $(m,n)$ in which $m$ $n$-agons meet at each vertex. Most of the time this tilings have to "live" in the hyperbolic plane. The edges of its polygons define a graph where two ...
7
votes
4answers
930 views
Area of a triangle $\propto\pi-\alpha-\beta-\gamma$
A hyperbolic geometry is a non-Euclidian geometry with constant negative curvature. It has the property that given a line and a point, many lines can be drawn containing the point that never meet the ...
7
votes
1answer
203 views
Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?
Motivation
I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points ...
7
votes
3answers
393 views
Simulation of Brownian Motion
If I want to simulate Brownian motion in the Euclidean space I can simulate it by a point that is moving a distance $\epsilon$ in an arbitrary direction then it randomly choose a new direction and ...
7
votes
2answers
133 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 ...
7
votes
3answers
597 views
how to generate tesselation cells using the Poincare disk model?
I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully.
I've been looking at M.C. Escher's ...
6
votes
2answers
695 views
Is an equilateral triangle the same as an equiangular triangle, in any geometry?
I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry?
I know they are the same in ...
6
votes
1answer
133 views
How to identify $SL(2,\mathbb{C})/SU(2)$ and the hyperbolic 3-space?
I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form
$$ g = \left( \begin{array}{cc}
\sqrt{t} & \frac{z}{\sqrt{t}}\\
0 & ...
6
votes
1answer
177 views
When does there exist an isometry that switches two subspaces?
Let $V$ be a real vector space of finite dimension and let
$\langle \cdot, \cdot \rangle$ be a non-degenerate symmetric
bilinear form on $V$. Let $U, W \subseteq V$ be linear
subspaces such that ...
5
votes
1answer
151 views
Parabolic elements correspond to punctures
In Mapping Class Group by Farb and Margalit page 22, they say:
Let $S$ be a hyperbolic surface. If a non-trivial element of $\pi_1(S)$ is represented by a loop (up to homotopy) around a puncture, ...
5
votes
1answer
90 views
Conformal automorphism of $H^n$
I was looking for the characterization ( or a complete list ) of the conformal automorphisms of the upper half space $H^n$ in $R^n$. I know that when $n=2$, it is $PSL(2,R)$ and when $n=3$, it is ...
5
votes
3answers
362 views
Expression of the Hyperbolic Distance in the Upper Half Plane
While looking for an expression of the hyperbolic distance in the Upper Half Plane $\mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\},$ I came across two different expressions. Both of them in Wikipedia.
...
5
votes
1answer
363 views
Shortest path on hyperboloid
On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
5
votes
2answers
115 views
Reflections generating isometry group
I was reading an article and it states that every isometry of the upper half plane model of the hyperbolic plane is a composition of reflections in hyperbolic lines, but does not seem to explain why ...
5
votes
1answer
345 views
Interpretation of Hyperbolic Metric and Möbius Transforms
I was wondering if someone could explain the interpretation of the following results. In hyperbolic geometry, we say that lengths are invariant under the action of Mob($\mathbb{H}$) if given any ...
5
votes
2answers
91 views
Two hyperbolic surfaces corresponding to conjugate Fuchsian groups are isometric
I have a basic question :
a) Suppose $\gamma $ and $ \gamma' $ be conjugate Fuchsian groups acting freely and properly discontinuously on the upper half-plane H to produce two Riemann surfaces $ ...
5
votes
1answer
56 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 ...
5
votes
1answer
413 views
Generalized Laws of Cosines and Sines
I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources)
(i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
5
votes
0answers
49 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 ...
4
votes
4answers
206 views
Completeness of Upper Half Plane
I am trying to prove that the upper half plane, defined as $\mathbb{H} = \{z \in \mathbb{C} : \Im(z)>0 \}$, is complete with respect to the hyperbolic metric.
First I note that if I have some ...
4
votes
1answer
99 views
Backslash notation: $\Gamma {\setminus} \mathbb{H}^n$
I encountered this notation in a paper by Carron:
When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ...
$\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My ...
4
votes
2answers
71 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 ...
4
votes
1answer
93 views
Fuchsian groups and surfaces
It's a fact that if two Fuchsian group are conjugate, the corresponding surfaces are isometric. Is the converse true ?
Take 2 isometric Riemann surfaces $S$ and $S'$(which are covered by the upper ...
4
votes
1answer
141 views
Hyperbolic triangle and two points in Poincare disk
Given a hyperbolic triangle $T$ and two points $p$ and $q$ in Poincare disk. Note that $p$ and $q$ are outside the triangle. If $p$ has shorter distances to the three vertices of $T$ than $q,$ can we ...
4
votes
1answer
175 views
Difference between a hyperbolic line and a geodesic
The setting for hyperbolic space in this question will be the upper half plane.
Now I know that to measure the distance between two points $p$ and $q$ in the upper half plane, we take
$ \inf ...
4
votes
2answers
559 views
Finding Möbius transformation from fixed points
Given a non-parabolic transformation which is also an orientation preserving isometry in the hyperbolic upper half plane union the boundary, if I know the two fixed points and they are two different ...
4
votes
2answers
227 views
How to analyze triangles in Lobachevsky geometry?
I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the ...
4
votes
1answer
395 views
Hyperbolic geometry. 3 dimensions. What is not well understood?
According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions.
http://mathworld.wolfram.com/HyperbolicGeometry.html
What isn't well understood about ...
4
votes
1answer
276 views
The law of sines in hyperbolic geometry
What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the ...
4
votes
1answer
37 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 ...
4
votes
1answer
107 views
Structure of $x^2 + xy + y^2 = z^2$ integer quadratic form
The pythagorean triples $x^2 + y^2 = z^2$ can be solved in integers using rational parameterization of solutions to $x^2 + y^2 = 1$.
It goes through $(1,0)$, then consider the line $y = -k (x - 1)$ ...
4
votes
1answer
155 views
Triangle inequality for hyperbolic distance
A quick way to define the hyperbolic metric in the Poincare disc is via the cross ratio: Given points a,b in the disc, let p,q be the endpoints of the hyperbolic line (halfcircle/line perpendicular to ...
4
votes
1answer
79 views
centralizers in hyperbolic manifolds are cyclic
I am having trouble seeing why this statement is true:
"If S admits a hyperbolic metric, then the centralizer of any non-trivial element of $\pi(S)$ is cyclic. In particular, $\pi(S)$ has trivial ...
4
votes
1answer
48 views
Can different uniformizations of Riemann surfaces be related somehow
Let $X$ be a hyperbolic compact connected Riemann surface. Let $U\subset X$ be an open subset. Assume that $U\neq X$.
We can uniformize $X$ by $\mathbf{H}$ directly to obtain it as a quotient of ...
4
votes
1answer
116 views
explicit isometry between metric spaces
Let $X=\mathbb{H}^2\times\left[0,1\right]$ (just $\left\{ (x,y,z)\big\vert y>0\right\}$ as a set) and consider the following two metrics: $$ds_1=\frac{dx+dy}{y}+dz$$ and ...
4
votes
1answer
237 views
Circle preserving homeomorphisms in the closure of $\mathbb{C}$ and Möbius Transformations
I am presently a learner of Hyperbolic Geometry and am using J. W. Anderson's book $Hyperbolic$ $Geometry$. Now the author presents a sketch proof of why every circle preserving homeomorphism in ...
4
votes
0answers
112 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
...
4
votes
1answer
158 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 ...
3
votes
6answers
199 views
Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?
I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...
3
votes
1answer
347 views
Hyperbolic metric on the torus?
Here is a silly mistake I am making: where exactly is the mistake?
I know that torus cannot hold a metric of constant curvature -1 ( hyperbolic metric ).
But what if I do this:
The upper ...
3
votes
2answers
424 views
Geodesic Uniqueness in the Hyperbolic Plane
I am studying Hyperbolic Geometry. At this part, I have proved that semicircles and straight lines orthogonals to the real axis are geodesics in the hyperbolic plane. But how I proof that this ...
3
votes
1answer
251 views
Models of hyperbolic geometry
Wikipedia states the following:
[The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
3
votes
1answer
169 views
Wikipedia article on Hyperbolic geometry
I was reading the Wikipedia article on hyperbolic geometry and have come across the line
geodesic paths are described by intersections with planes through the origin
Why is this necessarily ...
3
votes
2answers
263 views
Does anyone know a good hyperbolic geometry software program?
We are currently using this program called NonEuclid but it is a little frustrating to use sometimes and I was wondering if anyone knows another program for hyperbolic geometry.
3
votes
2answers
47 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 ...
