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

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108 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 ...
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73 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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199 views

Geometric interpretation of hyperbolic functions

When proving identities like $$\cosh(2x)=\cosh^2(x)+\sinh^2(x)$$ $$\cosh^2(x)=\sinh^2(x)+1$$ algebraically, I am beset by the feeling that there should be a geometrical interpretation that makes them ...
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351 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 ...
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77 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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143 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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56 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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141 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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76 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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18 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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91 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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300 views

Lines in coordinate system of Hyperbolic Plane

An orthogonal coordinate system of the hyperbolic plain can be set up by fixing an orgio $O$, an $x$-axis, a $y$-axis (intersecting each other at $O$ in angle $90^\circ$), and, from any point $P$ ...
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85 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
235 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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0answers
49 views

limit set of Kleinian grouos with closed manifolds as quotient

I'm trying to convince myself that if $M\cong\mathbb{H}^3/G$ is a closed hyperbolic 3-manifold then the limit set $\Lambda(G)$ equals the whole Riemann sphere $S_\infty^2$. My idea of the proof goes ...
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188 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces? In ...
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41 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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167 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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52 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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31 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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32 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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45 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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27 views

Big picture question about the Thurston Metric on Teichmuller Space

I'm having difficulty in appreciating the significance of the Thurston metric on Teichmuller space. It seems like a lot of work to develop the theory of this asymmetric metric space with fewer ...
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45 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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26 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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59 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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52 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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15 views

progression along geodesics

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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123 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 ...
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78 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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0answers
87 views

what is the use of the hyperboloid model for hyperbolic geometry?

I am quite new to hyperbolic geometry so even an answer that this question doesn't make any sense can be very helpful. As far as i understand: There are different models of a plane where hyperbolic ...
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15 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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26 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 ...
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43 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
210 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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0answers
37 views

Is $M_g$ a subvariety of $M_{h}$ for some $h>g$

Let $g\geq 24$. Then $M_g$ is of general type. Does there exist $h>g$ such that $M_g$ is a subvariety of $M_h$? That is, does there exist an immersion $M_g \to M_g$? If the answer is not known, ...
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184 views

Aristotle's Axiom in Hyperbolic Geometry

I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs ...
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172 views

Is there non-discrete group isomorphic to the fundamental group, what about the quotient?

It is known that (uniformization theorem) any Riemann surface can be written as the quotient of its universal cover by a discrete group (of Möbius transformations). This group is isomorphic to the ...
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69 views

Relations between Kleinian groups and quotient manifolds

In some specific situation there are some nice relations between a Kleinian group and its quotient manifold. For example, if $G$ is a once-punctured-torus group (i.e. a free subgroup of ...
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0answers
71 views

For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset

Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve. Let $P$ be the identity element of $E$. Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
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45 views

Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
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0answers
42 views

Why does $\mathrm{Aut}(\mathbb{H}) = \mathrm{Isom}^{+}(\mathbb{H})$?

Suppose $T \in \mathrm{Isom}^{+}(\mathbb{H})$ . With out loss of generality we may assume that $T$ fixes two points $P,P′$ on the imaginary axis $i\mathbb{R}$. Now let $Q \in \mathbb{H}$. Since ...
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0answers
133 views

Area in the upper half plane of a domain-Collar lemma

How can I compute the area of this region in the upper half plane $N=\{z\in U| 0<|z|<e^a,|\text{arg}(z)-\frac{\pi}{2}|<\theta_0, \text{Re}(z)\ge 0\}$, where $U$ denotes the upper half plane ...
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38 views

“Fundamental region” for non-discrete Moebius groups.

Suppose we are given a discrete, faithful representation $\rho$ of $F_2=\langle a,b|\rangle$, the free group on two generators, into $\mathbb{P}SL(2,\mathbb{R})$, so that the quotient is homeomorphic ...
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12 views

small deviations along hyperbolic G-orbit quasi-geodesics

Let $A$ and $B$ be two hyperbolic elements in $\mathbb{P}SL(2,\mathbb{R})$. If $p$ is a point in the hyperbolic plane, we can consider the broken geodesic ray $\gamma_A$ described by the vertices ...
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49 views

Internal angle formula of a generalized polygon as a function of side length and apothem

I am looking to compute the internal angle of a generalized regular polygon (spherical, euclidean, or hyperbolic) as a function of its apothem and side length. I know the equation for a euclidean ...
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46 views

Fundamental Domain for Congruence (mod 2) Group

How can I show that the area between the circles $|z|=1$, $|z+\frac{1}{2}|=\frac{1}{2}$, $|z-\frac{1}{2}|=\frac{1}{2}$ in the upper-half plane (here's a picture) is a fundamental domain for the ...
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32 views

How to get the smallest sample size with max probability in Hypergeometric Distribution

A body of students has 30 male students and 20 female students. Suppose a sample of n students are drawn from this population. What is the smallest n that can yield the maximum probability to have 5 ...
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67 views

Prove the Lobachevsky-Bolyai formula for the Klein model

I want to prove that e^(-d) = tan(Π(d)/d) in the Beltrami-Klein Model for the angle of parallelism in correspondence to the distance d, where d is the klein distance d(AB) = (1/2)|ln((AB,PQ)). A hint ...
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40 views

Equidistant Gergonne point in trebly asymptotic triangle, a consequence of Gergonne's Theorem in the Klein-Beltrami Model?

It is a theorem in hyperbolic geometry that inside every trebly asymptotic triangle (ABC) there is a unique Gergonne point G equidistant from all sides. Show that in the Beltrami-Klein model ...