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

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Why can't the pseudosphere be completed in $R^3$?

Without appealing to Hilbert's theorem on the non-embeddability of complete hyperbolic surfaces in $R^3$, is there a way to "see" that one can't extend the pseudosphere / surface of revolution of a ...
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107 views

Linear Isoperimetric Inequality is invariant under quasi isometry

Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality. My idea: Suppose $X$ satisfies a linear ...
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1answer
25 views

Why to include the $C$ in the formula for the distance in hyperbolic geometry?

I'm reading Penrose's: Road To Reality. First he gives the Lambert formula and later, he says that if you want, you can include the $C$ of the Lambert area formula. But It's not clear why I would ...
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Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
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1answer
81 views

How (not) to plot a Hyperbola? [closed]

I am trying to plot a Hyperbola in Wolfram Alpha. Its giving me a strange graph. How to correct that?
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2answers
72 views

Error when computing geodesics in hyperbolic half plane

It is known that the geodesic equations for the upper half plane equipped with the hyperbolic metric are $$x''=\frac{2x'y'}{y},$$ $$y''=\frac{(y')^2 -(x')^2}{y}.$$ It is also well known that the ...
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1answer
53 views

Geodesics in upper half-plane model of $\mathbb{H}$

On this page in Schlag's book on complex analysis, he is discussing the upper half-plane model of $\mathbb{H}^2$. He says for all $z_0\in \mathbb{H}$ $$\{T'(z_0) \mid T \in PSL(2, \mathbb{R}) ...
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1answer
101 views

Geodesic on hyperboloid and Poincare's Disk Model

I have two questions that 1. Why geodesic on hyperboloid corespond the arc in the Poincare's Disk Model? The hyperboloid : $x^2 + y^2 - z^2 = -1, \hspace{.15cm} z>0$ When any plane through ...
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1answer
271 views

Poincaré hyperbolic geodesics in half-plane and disc models

The objective of this post is to state that 1) the Poincaré hyperbolic metric results in a solution of complete geodesic circles in both half-plane and disk models. 2) the choice of one or other ...
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14 views

Cross Ratio of two rays through origin

There are two fixed and two variable concurrent rays of unit length in 3 space through the origin. How should the spherical coordinates of the two variable points be related to result in a constant ...
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1answer
31 views

The shape of the hyperbolic curves coordinates

Any one has an idea about hyperbolic coordinates ? and how to imagine it ? Indeed I am trying to find the shape of the coordinate curves far away from origin ! and what is the shape of them at $u=0 ...
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3answers
207 views

Distance formula for points in the Poincare half plane model on a “vertical geodesic”.

In comment at http://math.stackexchange.com/a/1381829/88985 at Distance of two hyperbolic lines is says (as i interpreted it) that the distance between two points $(a,r)$ and $(a, R)$ in the ...
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1answer
43 views

Equidistant points to a hyperbolic line

consider the Poincare upper half-plane model of hyperbolic plane $\mathbb{H}^2$ and a hyperbolic line $\ell\subset \mathbb{H}^2$ (or geodesic if you want). I would like to visualize the set of points ...
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1answer
44 views

Mapping from Poincare's disk model to UHP

I have a question that : How can I map any point in Poincare's disk model to Upper-half-plane model? I know the function $$f(z) = \frac{z + i}{iz+1}$$ But I want to know the geometric ...
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1answer
116 views

Proof of the ultraparallel theorem in the Beltrami Klein model

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct. (the ultra parallel theorem is ...
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1answer
50 views

On the hyperboloid model, if the point $\mathbf{v}$ gets translated to the origin, then where does the point $\mathbf{x}$ go?

Wikipedia has the answer in the case of the Poincaré disk model. When the point $\mathbf{v}$ is translated to the origin, then the point $\mathbf{x}$ is translated to $$\frac{(1 + 2\mathbf{v} \cdot ...
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2answers
155 views

Distance of two hyperbolic lines

Consider the upper-half plane model of the hyperbolic plane $\mathbb {H}^2.$ Now consider two lines in it given as $\ell_1:=\lbrace { (x, y)\in \mathbb {H}^2 \vert x^2 +y^2=r^2\rbrace}, ...
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1answer
95 views

Triangular Area on hyperbolic surface

I have read numerous paper over area calculation in hyperbolic geometry but just can't seem to understand how to calculate a triangle's area in hyperbolic geometry. It would be nice to have a proof ...
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65 views

Curvature of hyperbolic surface

So from what I understand the curvature of a surface by calculated by inversing the radius of the osculating circle. But if a hyperbolic surface have a negative curvature, wouldn't that imply the ...
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2answers
44 views

N-polygons in hyperbolic geometry

Let $N$ be an integer and we have two $N$-polygon $A_{1}A_{2}\ldots A_{N}$ and $A'_{1}A'_{2}\ldots A'_{N}$ such that the length of geodesic $A_{i}A_{i+1}$ is equal to the length of geodesic ...
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1answer
26 views

rotations and SU(1,1)

I'm interested in the isometries of the hyperbolic plane, i.e. the mappings which leave the geometric properties of objects invariant. I'm working with the Poincare disc model. My lecture notes ...
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61 views

Search for coverings in hyperbolic tessellations

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane? For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...
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1answer
33 views

What's the right way to calculate hyperbolic distance on the hyperboloid model?

I see in the Wikipedia article on the hyperboloid model and also in this other Math.SE question about the hyperboloid model that this is how you calculate distance on the hyperboloid model: Let $u = ...
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2answers
61 views

Isometric group of hyperbolic 3-dim manifolds

In the book : Foundation of Hyperbolic Manifolds There is a theorem that any finite subgroup of $ Isom(\mathbb{E}^n) $ fixes a point. And I hope to solve the following question : Any ...
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Corollary of Wolpert lemma

Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then ...
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2answers
36 views

What is the largest possible sum of all the angle measures of a $\Delta$ in hyperbolic space?

$\Delta ABC$ exists in hyperbolic geometry. What is the maximum value for $m\angle A+m\angle B+m\angle C$?
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31 views

Hyperbolic segment from $(0,0)$ to $(0,0)$

Can there be a segment on a hyperbolic plane that goes from point $(0,0)$ to $(0,0)$ in the hyperbolic plane. There are some rules, though for this to work: 1) The segment must apply to the rules of ...
2
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1answer
64 views

Heat kernel formula on hyperbolic plane well defined

Consider the heat kernel for the hyperbolic plane $\mathbb{H}^2$ and the corresponding heat kernel: $$k(x,y,t)=\frac{C}{t^{\frac{3}{2}}}\cdot ...
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1answer
42 views

Are bounded geodesics in the modular surface closed?

Let $M=\mathbb{H}/SL(2,\mathbb{Z})$ be the modular surface (which is noncompact but finite volume with the volume induced by the constant negative curvature metric inherited from $\mathbb{H}$). Any ...
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1answer
49 views

Finite hyperbolic geometry with ideal points

I was browsing "Thinking Geometricly: A Survey in Geometries" by Thomas Q. Sibley, 2015 and on page 388 it mentions a finite hyperbolic geometry of order 3 (3 points per line) consisting of 13 ...
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48 views

Equation of line in hyperbolic space

After a slightly peculiar dream the other night, I find myself suddenly inspired to do numerical simulations in three-dimensional hyperbolic space. For this to work, I need an equation of line in ...
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1answer
32 views

“Square,” line-preserving models of the hyperbolic plane

The Klein model of the hyperbolic plane is a line-preserving map from $H^2$ to the disk. Is there a model of the hyperbolic plane which is a line-preserving map from $H^2$ to $[0,1]^2$? By ...
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2answers
47 views

What are good references for the action of $\Gamma := \pi_1(S)$ on $S^1 = \partial \mathbb{H}^2$, where $S$ is a closed hyperbolic surface

To give some examples: what can we say about the action of $\Gamma$ on the set $V$ of points of $S^1$ that are not fixed for any element of $\Gamma$? Does there exist a Borel fundamental domain for ...
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1answer
77 views

Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
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52 views

Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & ...
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1answer
74 views

Existence of unique circle passing through interior points of unit disk meeting the boundary orthogonally

I am a self-studies and this is a hw problem from a complex analysis scourse I've been doing. The problem set pertains to the topic Automorphism Groups and has a high concentration of fractional ...
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2answers
148 views

Parallel transport of a vector in hyperbolic space, specifically in $\mathbb{H}$

Let us consider Poincaré's upper plane which is defined as $\mathbb{H} = \{ (x,y) | y>0\}$. This space has a Riemannian metric $g = \text{diag}(1/y^2, 1/y^2)$. Now let us consider a differential ...
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137 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 ...
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1answer
193 views

Mapping the Poincare disk model to the Poincare half plane model

I am puzzeling with the following: Given a point $ A = ( a_x, a_y) : a_x^2+ a_y^2 \le 1 $ in the Poincare Disk model ( https://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model ) to which point does ...
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1answer
73 views

A lift of isometry to universal covering

Let $M$ be a compact Riemannian manifold, $\bar M \to M$ be its universal covering and $\phi \in Isom(M)$ be an isometry of $M$. Is it true that, if $\phi$ is isotopic to the identity map of $M$, than ...
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43 views

Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group ...
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1answer
72 views

Edge length of hyperbolic tesselations

If I have a general uniform tesselation in hyperbolic plane (same configuration of regular polygons at every vertex, but multiple types of polygons allowed), how can I find the edge length and/or ...
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20 views

Hyperbolic length does not depend on the subdivision.

I'm reading some notes on hyperbolic surfaces by François Labourie and there's an exercise I can't figure out. I have to prove that the length l(c) of a curve does not depend on the subdivision. It's ...
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1answer
192 views

Hyperbolic geometry and the Triangle Inequality

In Is the shortest path in flat hyperbolic space straight relative to Euclidean space? I answered by refering to the Triangle Inequality (https://en.wikipedia.org/wiki/Triangle_inequality , Euclid's ...
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1answer
34 views

Relationship between Hyperboloid model of hyperbolic space and disc model / confused by a picture.

I am confused by this picture: https://en.wikipedia.org/wiki/File:HyperboloidProjection.png What is wrong with projecting from the origin, and using the disc at $t = 1$? After doing some computation ...
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214 views

Classification of conics in hyperbolic plane

How many different types of conics exist in hyperbolic plane? Euclidean geometry has three, of course. But when I was trying to find out results for the hyperbolic plane, the best thing I found ...
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3answers
127 views

how to construct a line on a poincare disk?

Given the Euclidean coordinates of two points (p1, p2) and (q1, q2) in the unit circle, how do I construct the Euclidean circle x^2 + y^2 + fx + gy+1 representing the hyperbolic d-line on the poincare ...
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2answers
109 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
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1answer
21 views

Whether the hyperbolic trigonometry rule holds for all models

I am reading a paper concerning hyperbolic geometry. It represents some results like the hyperbolic cosine rule. Consider a hyperbolic triangle with side lengths $a$, $b$, $c$ and angles $\alpha$, ...
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90 views

A triangle in the Poincare disc model

Suppose that we have a triangle in the Poincare disc model such that the internal angels are all equal. Then Does it imply that the lengths of sides are all equal? By length of a side, I mean the ...