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

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2
votes
1answer
316 views

Proving an equality for an equilateral triangle in the Poincare model

I've been working a good while trying to establish an equality, but have made little success. Suppose you're working in the Poincare disk model inside an ambient Euclidean plane. If an equilateral ...
4
votes
4answers
258 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 ...
10
votes
2answers
190 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 ...
3
votes
1answer
162 views

Möbius Transforms that preserve $\mathbb{H}$

I know that every möbius transform that preserves the upper half plane is of the form $m(z) = \frac{az+b}{cz+d}$, where $a,b,c,d \in \mathbb{R}$, or $m(z) = \frac{a\bar{z} + b}{c\bar{z} + d}$, where ...
4
votes
1answer
1k views

Möbius Transforms that preserve the unit disk

Say I wish to prove that every möbius transformation of the unit disk onto itself can be written in the form $A(z) = e^{i\theta}\frac{z+a}{1+\bar{a}z}$, where $\theta$ is a real number and $a$ is a ...
1
vote
1answer
643 views

Deriving the distance between two distinct points on the Upper Half Plane $\mathbb{H}$

I am trying to derive the distance between two arbitrary points in hyperbolic space; the model I'm using is the upper half plane model. So the distance is just $\int_f \rho(z) dz$, where $\rho(z) = ...
5
votes
1answer
496 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 ...
0
votes
3answers
371 views

Exciting Topics in Hyperbolic Geometry

I am a first year student and a learner of hyperbolic geometry. I was wondering if you could suggest some exciting topics to research about in this field (some people suggested fundamental polygons ...
8
votes
3answers
491 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 ...
1
vote
1answer
262 views

Hyperbolic area and $SL_2$

Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that: a. The measure $\mu$ is invariant under all $g \in ...
4
votes
1answer
277 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 ...
10
votes
1answer
288 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 $\geq 2$) ...
3
votes
1answer
486 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 ...
0
votes
1answer
96 views

Question on proof in “Primer on MCGs”

This is a question about the proof of Proposition 1.4 in Farb and Margalit's "Primer on Mapping Class Groups" (in v. 5.0, it is on page 37 in the PDF, which you can download here). The proposition ...
1
vote
1answer
112 views

What is the cardinality of a subset of the hyperbolic upper half plane?

Given a subset of the hyperbolic upper half plane, say an ideal triangle (so with vertices on the boundary), what is the cardinality of all points contained in the interior?
4
votes
2answers
715 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 ...
5
votes
2answers
112 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 $ ...
4
votes
2answers
263 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 ...
12
votes
3answers
505 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, ...
1
vote
1answer
259 views

Geodesic on half-plane determined by tangent vector

The upper-half plane $\mathbb H$ carries a hyperbolic metric and the geodesics are semicircles with base on the real line. We consider oriented geodesics. Let $x \in \mathbb H$ and let $v$ be a unit ...
4
votes
1answer
546 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 ...
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 ...
88
votes
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. ...