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

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0
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1answer
69 views

Poincare disc model problem. find $d_h(A,B)$

Consider $\triangle ABC$ on a poincare disc. On $\triangle ABC$, $\angle C=90^\circ$, $d_h(B,C)=a$ and $d_h(A,C)=b$ In this situation, find $d_h(A,B)$. I'm taking a course but I cannot follow ...
1
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1answer
326 views

prove that the sum of the angles in any triangle is less than 180 in hyperbolic geometry (or poincare model).

We could use poincare disc model as a hyperbolic geometry model. I have difficulty understanding poincare disc model. So is there someone to help?
1
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1answer
361 views

Convert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels $x$ units in Euclidean space, how ...
2
votes
1answer
361 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
2
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1answer
277 views

Distance in hyperbolic geometry

In Euclidean geometry, we have that the distance between two points $p$ and $q$ in $\Re^n$ is $\sqrt{(p_1^2-q_1^2) + (p_2^2-q_2^2) + \ldots + (p_n^2-q_n^2) }$ (if we denote the points by $p = (p_1, ...
3
votes
1answer
238 views

Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?

I read the answers to this very interesting question and saw that we can in fact embed the Euclidean plane into hyperbolic 3-space using what is called a horosphere. However, as Hilbert showed us, the ...
6
votes
1answer
981 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 ...
0
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1answer
67 views

What would be called as the shape of $xy=10$ in 3-dimensional space?

As title says, what would be called as the shape of $xy=10$ in 3-dimensional space? It doesn't seem to be paraboloid nor hyperboloid...
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0answers
27 views

Proof: Every H-Projection is product of H-Reflections

I have some problems with the proof of the statement above (sorry for a bad translation I guess). First of all some definitions: Definition 1 (möbius-transformation): Let $A \in \mathbb{C}^{2 ...
3
votes
1answer
164 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
1
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0answers
52 views

JSJ-decomposition of a non-hyperbolic 3-manifold

Suppose $M$ is a $3$-manifold. Then you can split it over spheres. This is the "prime decomposition" and is unique. You can then split the components of this decomposition along tori. If you leave the ...
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0answers
50 views

boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on ...
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5answers
82 views

Easy explanation of non-abelianness of hyperbolic curves

I'm looking for easy proofs (or just an easy proof) of the following statement: Let X be a hyperbolic Riemann surface, i.e., $X$ is a Riemann surface and the universal covering of $X$ is the complex ...
5
votes
1answer
975 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 ...
2
votes
1answer
155 views

Seifert manifolds and Fuchsian group

A Fuchsian group is a discrete subgroup of $PSL(2, \mathbb{R})$. Let $M$ be a Seifert manifold (maybe with boundary) and $t \in \pi_1(M)$ the class of a regular Seifert fiber. Hempel claims in his ...
1
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1answer
296 views

Questions about triangles, n-gons, and tessellations of the hyperbolic plane

Why there are infinitely many regular tessellations of the hyperbolic plane? Can there be a triangle made up of three straight lines in the hyperbolic plane? I know it's impossible since the angle ...
0
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1answer
91 views

What is complex conjugation in Hyperbolic Geometry

Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite ...
1
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1answer
138 views

justifying reflection across line in beltrami-klein model

Justify the following construction of the Klein reflection A' of A across m. Let Λ be an end of m and P be the pole of m. Join Λ to A and let this line cut y (which is the circle, my note) ...
2
votes
1answer
69 views

Simple closed geodesic around two hyperbolic cusps.

Consider a connected hyperbolic $2$-manifold $M$ with cusps. Consider a simple closed geodesic in $M$, which dissects $M$ into two components. Assume that one of the components contains exactly two ...
1
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1answer
90 views

projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
0
votes
1answer
45 views

Verifying the vertices of a Hyperbola

In this exercise it is required to verify that the points O & A(4,0) are the vertices of the Hyperbola H , as you can see in the marked part. Can someone help verify that ?
4
votes
1answer
79 views

Computing the volume of a hyperbolic knot

Could anyone show me or refer me to a link where the volume of a hyperbolic knot, say, the figure-8 knot, is computed (well, in fact estimated) explicitly and not only having the procedures outlined?
2
votes
1answer
97 views

Quotient under a Fuchsian Group

Let $D$ be the unit disk in the complex plane, and let $H$ be a fuchsian group generated by one fixed point free element, say $a$. What it the quotient $D/H$? Attempt: The quotient is biholomorphic ...
2
votes
1answer
98 views

A Fuchsian Group?

Let $p_k := e^{\pi/2 i k}$, $k \in \{0, 1,2,3\}$. Let $b_k$ the geodesic of the hyperbolic disk connecting $p_k$ and $p_{k+1(\text{mod}4)}$. For instance, $p_0$ and $p_1$ are connected by the lower ...
2
votes
2answers
190 views

Hyperbolic Geometry - reference request [duplicate]

I need some information about Hyperbolic Geometry. For example, Spherical Geometry is a subsection of Hyperbolic Geometry or no? Can you suggest to me a book or some other reference to help me ...
3
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0answers
86 views

Question regarding the projective models of the anti-de-Sitter spaces and good online references for learning them from the scratch? (Specifics below)

As my title says above, I am trying to find answers to and also good online reference where I can find complete description of projective models of hyperbolic space, de-Sitter space and anti-de-Sitter ...
2
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1answer
162 views

Isometries of a hyperbolic quadratic form

I am reading an article that says "The group of isometries (of a hyperbolic space) of a hyperbolic quadratic form in two variables is isomorphic to the semi-direct product $\mathbb{R} \rtimes ...
2
votes
1answer
171 views

Isometry fixing two points of a geodesic line

Let $H$ be a hyperbolic space, and let $\Gamma \subset H$ be a geodesic line, i.e., the image of an isometry from $\mathbb{R}$ to $H$. If $f$ is an isometry of $H$ that fixes two distinct points of ...
2
votes
1answer
204 views

Arc length parameter s

Consider the metric $$ds^2 = \frac{dx^2+dy^2}{y^2}.$$ Assume $R>0, a\in\mathbb{R}$. Consider the curve $$\gamma(\theta)=(a+R\sin\theta,R\cos\theta)$$ for $-\frac{\pi}{2}\leq\theta\leq ...
2
votes
2answers
234 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
0
votes
1answer
124 views

Möbius transformation that preserve distance for two pair of given points in $\mathbb{H}$.

I need to prove that for a given two pair of points $(z_1,z_2)$ and $(w_1,w_2)$ in $\mathbb{H}$ (Poincaré's upper half plane), where $d_{\mathbb{H}}(z_1,z_2)=d_{\mathbb{H}}(w_1,w_2)$, there is an ...
4
votes
1answer
387 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 ...
0
votes
2answers
314 views

circle reflections in hyperbolic geometry

Determine the equation of the circle reflection of the circle $x^2 + y^2 = 1$ if the circle of reflection is $x^2 + y^2 + 2x = 0$. I'm learning about circle inversion but I still don't get what this ...
2
votes
1answer
50 views

Need help with finding length of sides and angles of a triangle in upper half plane model

The given points are $i, 3i, 1 + 2i$ I know that the distance for points on a vertical line can be found by using the formula $$\ln\left|\frac{y_2}{y_1}\right|$$ So the distance between points $ i$ ...
1
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1answer
283 views

Finding an angle of a triangle in the upper half plane model given three points

I've been given three points in the upper half plane $(i, 3i, 1 + 2i$), and one of the homework questions that asked is to find the angles of the given triangle. A previous problem asks to find the ...
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0answers
198 views

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides?

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides? I have been trying to visually see if that is possible or not.
1
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1answer
69 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?
1
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1answer
105 views

hyperbolic quadrilateral angles

On the hyperbolic plane, if I have a quadrilateral that has all congruent interior angles $\alpha$, how do I figure out what $\alpha$ is? I know in Euclidean geometry one could just use ...
1
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1answer
124 views

geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
1
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1answer
133 views

How does my Beltrami-Klein model look?

Did I sketch the picture right based off of the specific instructions given in the problem?
1
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2answers
225 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.
2
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2answers
49 views

Hyperbolic quadrilaterals with two adjacent right angles

For convenience we'll work in the hyperbolic upper half plane $H$. We are given a hyperbolic quadrilateral $Q$ with vertices $a,b,c,d$ and geodesic segment edges $[ a,b ]$ $[ b,c ]$ $[ c,d ]$ $[ d,a ...
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0answers
308 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
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1answer
62 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 ...
3
votes
2answers
151 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 ...
1
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1answer
92 views

Projecting external points to a circle: Distance order preserving?

Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation: I compute the point of intersection of the i) circle and the ii) line joining each ...
8
votes
2answers
317 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
1answer
112 views

Any hyperbolic $n$-simplex is contained in an ideal simplex

Recall that an $n$-simplex in $\overline{\mathbb{H}^n}$ (the closure of $n$-dim hyperbolic space) with vertices $v_0,...,v_n\in \overline{\mathbb{H}^n}$ is the closed subset of $\mathbb{H}^n$ bounded ...
1
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1answer
259 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 ...
3
votes
2answers
160 views

topic for presenting in hyperbolic geometry

For my course work, i have to give a presentation of 20-30 min presentation in hyperbolic geometry. Can any one suggest some topic(or any interesting theorem) in this area.I want to present some thing ...