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

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96
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4answers
2k views

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. ...
26
votes
11answers
6k 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 ...
21
votes
1answer
667 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 ...
15
votes
3answers
553 views

How did Beltrami show the consistency of hyperbolic geometry in his 1868 papers?

This is in response to comments and the answer by user studiosus to this question: As for Beltrami's work: Consistency of a geometry from (post) Hilbert viewpoint has nothing to do with ...
14
votes
3answers
2k 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 ...
12
votes
3answers
539 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, ...
12
votes
2answers
205 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 ...
11
votes
1answer
245 views

Law of sines: uniform proof of Euclidean, spherical & hyperbolic cases

There is a unified formulation of law of sines which is true in all 3 constant curvature geometries (Euclidean, spherical, hyperbolic): $$ \frac{l(a)}{\sin\alpha}= \frac{l(b)}{\sin\beta}= ...
10
votes
1answer
338 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 ...
10
votes
1answer
347 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 $\geq 2$) ...
8
votes
2answers
88 views

Totally geodesic hypersurface in compact hyperbolic manifold

In [Zeghib: Laminations et hypersurfaces géodésiques des variétés hyperboliques, Annales scientifiques de l'ENS, 1991] it is shown, that in a compact manifold of negative curvature, there exists only ...
8
votes
4answers
987 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 ...
8
votes
3answers
558 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 ...
8
votes
2answers
325 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 ...
8
votes
1answer
153 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 ...
8
votes
1answer
72 views

constant-curvature Riemannian metric for Bring's surface

There is a well-known and very symmetric space that is called either "Bring's curve" or "Bring's surface", depending upon the context. (Bring was a Swedish mathematician in the 18th century.) Let's ...
8
votes
1answer
130 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
7
votes
4answers
1k 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
2answers
3k 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 ...
7
votes
2answers
108 views

What hyperbolic space *really* looks like

There are several models of hyperbolic space that are embedded in Euclidean space. For example, the following image depicts the Beltrami-Klein model of a hyperbolic plane: where geodesics are ...
7
votes
2answers
449 views

Symbolic coordinates for a hyperbolic grid?

Rephrasing     (one year later)    (original question is below) Apparently the original question wasn't clear, or nobody knows an answer (or both). So I will try to rephrase it. Look at your ...
6
votes
1answer
275 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, ...
6
votes
1answer
462 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 ...
6
votes
1answer
190 views

Curvature of De Sitter's space: where does the sign comes?

Consider $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, where: $${\rm d}s^2 = {\rm d}x^2 + {\rm d}y^2 - {\rm d}z^2.$$ We have both the hyperbolic space: $$\Bbb H^2(-1) = \{(x,y,z) \in \Bbb L^3 \mid ...
6
votes
1answer
1k views

What is the relationship between hyperbolic geometry and Einstein's special relativity?

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic ...
6
votes
1answer
105 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 ...
6
votes
1answer
332 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
2answers
37 views

Is it possible to distinguish rest and movement in hyperbolic universe?

Imagine a large body (for example, a planet) in 3D hyperbolic space. Now imagine the planet starts moving in a straight line at constant speed. In Euclidean space, all points would move along ...
6
votes
1answer
214 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 ...
6
votes
2answers
624 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
5
votes
2answers
682 views

Creating a Hyperbola with a Flashlight

I ran into this problem in a textbook and was intrigued by it. Conics are generally formed through different cuts one can make with the shape of a cone. But, there have been recent discussions on ...
5
votes
2answers
152 views

An axiomatic treatment of hyperbolic trigonometry?

I would like to see results derived in hyperbolic trigonometry synthetically, i.e. just by working from axioms, for example the ones given by Hilbert (or even Tarski). Most authors seem to discuss ...
5
votes
3answers
2k views

The shape of Pringles potato chip

Why the shape of Pringles potato chip is hyperbolic paraboloid? I found several articles that say the shape is hyperbolic paraboloid, but cannot find out why it is so. Does anyone have reasonable ...
5
votes
3answers
1k 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
216 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
1answer
662 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 ...
5
votes
2answers
152 views

Relation: Modular Forms and hyperbolic geometry, or, why do they map from $\mathbb{H}$?

In my very young mathematical career, I have worked a lot with modular forms. Recently, I worked as a teaching assistant in a course about geometry. At the end of the course, we dealt with hyperbolic ...
5
votes
2answers
128 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
1k 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 ...
5
votes
1answer
60 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 ...
5
votes
2answers
247 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
2answers
499 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 ...
5
votes
1answer
582 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
1answer
107 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
2answers
81 views

An interesting point of a triangle. (Help needed to prove a statement.)

Consider a triangle whose sides are segments of $\color{red}{\text{line}}$, $\color{blue}{\text{line}}$, $\color{green}{\text{line}}$ falling in the circum-circle $c$. Let ...
5
votes
1answer
606 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
65 views

Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or ...
5
votes
1answer
89 views

Hyperbolic metric spaces

I'm trying to prove a simple proposition wich is in Burago's "A Course in Metric Spaces" (Exercise $8.4.5$, p.$287$). Before exposing my problem, let me give some definitions. A metric space $(X,d)$ ...
5
votes
0answers
169 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
5
votes
0answers
106 views

Tilings of the Hyperbolic plane

Given a tiling of the hyperbolic plane projected onto a unit disc such as this which can be considered as a graph. I then define some functions: $f(r) =$ number of graph nodes contained within the a ...