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

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5
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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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169 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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106 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 ...
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519 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 ...
3
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40 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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61 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 ...
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43 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 ...
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88 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 ...
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170 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 ...
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57 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 ...
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174 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 ...
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327 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 ...
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88 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 ...
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194 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 ...
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24 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 ...
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32 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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43 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 ...
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51 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
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25 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 ...
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47 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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56 views

Fundamental solution of Poisson equation in the Hyperbolic Plane

If we consider the Poisson's equation $$ -\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n, $$ we can construct the fundamental solution $$ u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy, $$ where $\Gamma$ is ...
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56 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 ...
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55 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 ...
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116 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 ...
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22 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 ...
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95 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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59 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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77 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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17 views

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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48 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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23 views

Scale and the models of the hyperbolic plane

I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical) So at the boundary line the scale is $ 0 $ or $ ( 1 : \infty ) $. at the ...
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27 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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20 views

Length of a hypercycle.

I was a bit puzzeling about what is the length of a hypercycle, horocycle and the line segment between two points. and found out that if $h$ is the length of one of the two horocycles between P and ...
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43 views

Find the hyperbolic length of the geodesic segment

I'm reading my textbook and I'm trying to make sense of this example. So the place with the red star shows the actual process of calculating the hyperbolic length. My question is how they get the ...
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58 views

We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2) $$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2) $$ And in the hyperbolic ...
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44 views

Comparing two fundamental domains for $\Gamma(2)$

(0). My question concerns the relation between two different fundamental domains for the group $$ \Gamma(2)= \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix}\in SL_2(\mathbb Z) \; ...
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79 views

Mapping a line in the hyperboloid model of $\mathbb{H}^2$ to a circle in the Poincaré model

In the hyperboloid model of $\mathbb{H}^2$ a point $P(x,y,z)$ is the intersection of the vector $(x,y,z)$ with the upper sheet of the hyperboloid $x^2 + y^2 - z^2 = -1$, and a line $L(a,b,c)$ is the ...
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42 views

Side and angle relations in a hyperbolic quadrilateral.

Let $PQRS$ be hyperbolic quadrilateral, i.e. a quadrilateral in $\mathbb{H}$ whose sides are hyperbolic geodesic. Let length$(PQ)=l_1$, and length$(PS)=l_2.$ Also $\angle SPQ=\theta_1$, $\angle ...
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94 views

measuring the curvature of a pringle

Assuming for the sake of argument that a pringle shaped potato chip has a constant negative curvature, (which according to http://math.stackexchange.com/a/617610/88985 it is not ) see also picture ...
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75 views

Hyperbolic vs Euclidean Brownian Motion

In this article, page 4 of the linked pdf file, Lalley and Sellke claim that a hyperbolic Brownian motion can be obtained by time-changing a 2-dimensional Euclidean Brownian motion, conditioned to ...
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135 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' ...
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45 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 : ...
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39 views

the fundamental group acts on half upper plan

Let $S$ be a compact oriental surface without boundary of genus $g\ge 2$, then its universal covering is $\mathbb{H}^2$, I am confused with 2 facts following: (1) $\rho:\pi_1(S)\hookrightarrow ...
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71 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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69 views

Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies ...
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33 views

Does a lattice in $PSL(2,\mathbb{R})$ stabilizing $\infty$ have a domain with vertex at $\infty$?

Suppose $\Gamma$ is a lattice in $PSL(2, \mathbb{R})$ acting on the upper half plane. Suppose that the stabilizer in $\Gamma$ of the point at infinity is nontrivial. Does it then follow that the ...
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98 views

The distance between two distinct points in the upper half plane

I'm trying to derive the distance between two distinct points in hyperbolic space and I'm working on the upper half plane. So, with the parametrization $\sigma(t): x=r\cos(t), y=r\sin(t),\; ...
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84 views

References for Hyperbolic Graph Theory

I'm sorry to disturb you but I really got stuck! I can't find any clear and, somewhat, complete reference for this topic. I'm looking for a book, or review, or survey or course notes regarding ...
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24 views

Progression of a point along geodesics under the action of hyperbolic Möbius transformations

Suppose that $X$ and $Y$ are two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$ with axes intersecting at, say, the centre $O$ of the hyperbolic disc model. Suppose also that the angle between ...
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266 views

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...