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

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5answers
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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 ...
4
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1answer
382 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
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1answer
98 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 ...
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1answer
142 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 ...
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0answers
47 views

Drawing graphics on a pseudosphere

I'm pretty sure this question is going to be very difficult to answer, so I will do my best to explain my problem, and maybe, just maybe someone will have a good answer. I have written a program ...
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1answer
71 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 ...
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1answer
95 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) ...
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1answer
44 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 ...
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1answer
70 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 ...
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1answer
42 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
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1answer
55 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
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1answer
79 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
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1answer
67 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 ...
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2answers
122 views

Hyperbolic Geometry - reference request

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
73 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
91 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
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1answer
117 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
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1answer
184 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
131 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 ...
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1answer
80 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 ...
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1answer
55 views

Help with hyperbolic geometry problem

construct two lines through the point $(3,1)$ that are parallel to the line $x=7$ construct two lines through the point $(3,1)$ that are parallel to the line $x^2 + y^2=36$
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1answer
148 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 ...
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1answer
169 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 ...
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1answer
40 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$ ...
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1answer
163 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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1answer
79 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.
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1answer
45 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?
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1answer
80 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 ...
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1answer
85 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 ...
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1answer
98 views

How does my Beltrami-Klein model look?

Did I sketch the picture right based off of the specific instructions given in the problem?
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1answer
116 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.
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2answers
28 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
201 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 ...
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1answer
59 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 ...
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2answers
125 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 ...
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1answer
70 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 ...
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1answer
34 views

conformal mapping

Prove that application ζ^-1:B^2---> H^2 is conform. ζ^-1(x1,x2,x3)=(x1/1+x3 , x2/1+x3) B^2 is unit disc and H^2 is hyperbolic 2-space and conformal mapping preserve angles.
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2answers
195 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
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1answer
63 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 ...
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0answers
200 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
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2answers
131 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 ...
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1answer
83 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 ...
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1answer
81 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
3
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1answer
77 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 ...
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1answer
135 views

Length of a curve on $S^2$

$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$? $2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates? ...
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1answer
49 views

Questions on two elements in a Fuchsian group which have at least one common fixed point

This is a homework question I am unable to solve. Let $A,B \in PSL(2,R)=Aut(H)$. Assume none of them are elliptic and they have : Case 1) one common fixed point at the boundary of $H$ (i.e. they ...
3
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1answer
177 views

Set of points equidistant from two points in hyperbolic space.

Given two points $p,q$ in the hyperbolic plane, show that the set of points equidistant from $p$ and $q$ is a hyperbolic line. I am unsure how to proceed with this question. Would it be easier to use ...
3
votes
2answers
178 views

Reflection in a hyperbolic line formula

Let $H$ denote the upper half-plane model of hyperbolic space. If $L$ is the hyperbolic line in $H$ given by a Euclidean semicircle with centre $a\in \mathbb{R}$ and radius $r >0$, show that ...
3
votes
2answers
128 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 ...
5
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1answer
230 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 ...