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

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6
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102 views

How to parameterize these pretty hyperbolic (Amsler) surfaces?

I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature. To this end, I attempted to model the ...
5
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77 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
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193 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
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117 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 ...
4
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33 views

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
4
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45 views

Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...
4
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644 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric ...
4
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199 views

How do we define a complete metric on a Riemann surface with punctures?

This question is related to another question. If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this ...
3
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0answers
107 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers ...
3
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45 views

What are some good graphics programs for depicting hyperbolic geodesics?

I'm looking for some software that allows one to draw accurate pictures in hyperbolic space. In particular, I want to be able to specify pairs of points and generate the geodesic between them, in ...
3
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0answers
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) & ...
3
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65 views

Parallel postulate in hyperbolic geometry question

Let $L =\{y=0\}\cap\mathcal{H}^{2}$, and let $P=(3,2,2)$. Show that the parallel postulate fails in $\mathcal{H}^{2}$ by giving two lines $L',L'' \in \mathcal{H}^{2}$ with $P\in L',L''$ and $L\cap ...
3
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51 views

Finding a fundamental polygon for two-generator subgroup of PSL(2,R)

Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like ...
3
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99 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 ...
3
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0answers
182 views

Axis of the product of two loxodromic isometries

Suppose that $X$ and $Y$ are two loxodromic isometries of the hyperbolic space and that the product $XY$ is also a loxodromic element. We consider the axes of these three elements. I'd like to know if ...
3
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0answers
58 views

reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result: For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
3
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0answers
481 views

the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
3
votes
0answers
89 views

Construct tiling group from hyperbolic polygon

Given a hyperbolic $4n$-gon $P$ in the Poincaré disk, how can we construct explicitly the subgroup $G < \mathrm{Aut}{\left(\mathbb{D}\right)}$ which gives a tiling of $\mathbb D$ with fundamental ...
3
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0answers
202 views

The fundamental group of the mapping torus is doubly degenerate

Consider an hyperbolic compact surface $S$ (hence with genus $>1$) and a Pseudo-Anosov diffeomorphism $\varphi\colon S\to S$. We call "mapping torus" the 3-manifold ...
2
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40 views

Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
2
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0answers
23 views

congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
2
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0answers
31 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
2
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0answers
34 views

another pythagorean theorem in hyperbolic geometry

on https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry it says However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition ...
2
votes
0answers
63 views

Villarceau circle as a Loxodrome

A circular Clifford torus (radius at flat circle = h, section radius $ a , a<h $ ) is cut by a plane at an angle $ \cos \alpha = a/h \tag{1} $ centrally to the symmetry axis, the line of ...
2
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0answers
37 views

Example of a doubly degenerate Kleinian group which does not come from a mapping torus

Doubly degenerate Kleinian groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as ...
2
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0answers
104 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 ...
2
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0answers
33 views

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 ...
2
votes
0answers
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 ...
2
votes
0answers
76 views

constructing an equilateral triangle in the Beltrami klein model

I am puzzeling with the following: Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model) and given ...
2
votes
0answers
77 views

Calculating hyperbolic length

So, I am looking at a question and I'm having a hard time solving it. So I know the formula but my question is first, what is $\alpha$ and what is $\beta$? So I calculated some values: I found ...
2
votes
0answers
39 views

Compute hyperbolic length of the arc of the circle

Compute the hyperbolic length of the arc of the circle $ x^2 + y^2 = 25$ that lies between (3, 4) and (4, 3). From my notes I know the formula is $$ \ln \frac{{\csc \beta - \cot \beta }}{{\csc ...
2
votes
0answers
61 views

Thread constructions in the Poincaré's disc

I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc ...
2
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0answers
58 views

Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that ...
2
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0answers
222 views

What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
2
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0answers
50 views

Submanifold of a Kobayashi hyperbolic manifold

Let $M$ be complex manifold which is Kobayashi hyperbolic. Let $N$ be a submanifold of $M$ obtained as the zeroes of an analytic submersion $f : M \rightarrow R$, $R$ complex manifold. Question : ...
2
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0answers
66 views

Big picture question about the Thurston Metric on Teichmüller Space

I'm having difficulty in appreciating the significance of the Thurston metric on Teichmüller space. It seems like a lot of work to develop the theory of this asymmetric metric space with fewer ...
2
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0answers
132 views

What is the Area formed when a line is traced between two 3D curves?

This question is quite related to intersection of cylinders, Hyperbolic paraboloid and modelling. I am welding a trunnion to a pipe (both are hollow cylinders in different geometry). They intersect ...
2
votes
0answers
24 views

software to decide whether a 2-generator subgroup of PSL(2,R) is discrete/free

Gilman developed an algorithm with polynomial complexity that, given two elements in PSL(2,R), decides whether the group they generate is free/discrete or not. I was wondering whether anybody ever ...
2
votes
0answers
97 views

Distance realizing geodesic in a hyperbolic surface

Suppose $S$ is a hyperbolic surface with geodesic boundary and $P$ is a hyperbolic pair of pants with $a$, $b$, $c$ geodesic boundary. Let $\gamma$ be the unique geodesic realizing the distance ...
2
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0answers
64 views

ruling out non Pseudo-Anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
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0answers
17 views

The triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$

This answer made me wonder if there is a geometrical way to prove that the triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$. In other words, how can we ...
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0answers
32 views

Find the curvature of the metric $ds=\frac{|dz|}{(1+|z|^2)}$ on $\mathbb{C}$

The curvature of the metric $g$ is defined as $$k(z)=-\bigg(\frac{2}{\alpha(z)}\bigg)^2 \partial \bar\partial log \alpha(z)$$ where $\alpha$ is positive and real valued. Also ...
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0answers
30 views

Does anything obstruct Mostow-Prasad rigidity for orbifolds?

If we phrase the Mostow-Prasad rigidity theorem algebraically, it goes like this (let $\mathcal{H}^n$ be a model for hyperbolic $n$-space). For $n>2$: if ...
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32 views

Fundamental domain of a Fuchsian group is non-compact if the group contains parabolic element

Suppose a discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ acts on $\mathbb{H}^2$. Why is the fundamental domain non-compact if $\Gamma$ contains a parabolic element? Thanks in advance.
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56 views

Kobayashi distance on the Siegel upper half space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article "inavariant metrics on ...
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41 views

Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the Poincaré ball model is $$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian ...
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0answers
58 views

spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
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0answers
43 views

Hyperbolic Geometry - Parabolic Matrix?

In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ...
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0answers
57 views

Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
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47 views

Geodesics in a hyperbolic plane like space

For $|\rho| < 1$ and $\sigma >0$ consider the Riemannian metric \begin{equation} g:= \begin{pmatrix} \frac{1}{\left( 1-\rho^2 \right)y^2} & \frac{-\rho}{\sigma\left( 1-\rho^2 \right)y^2} \\ ...