The hyperbolic-geometry tag has no usage guidance.
2
votes
0answers
98 views
Gauss Bonnet theorem calculation for pseudosphere
In an attempt to verify Gauss-Bonnet theorem for a Beltrami pseudosphere I calculated a simple case of the Riemann sphere. Taking curved radius $a$ for geodesic polar co-ordinate from the smooth ...
5
votes
1answer
121 views
Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces
I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using ...
10
votes
0answers
183 views
Why are the medians of a triangle concurrent? In absolute geometry
This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...
13
votes
1answer
80 views
Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?
Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
What if in Question 1 we restrict the cone angles to be $...
3
votes
2answers
101 views
Simple Closed Hyperbolic Geodesics on Punctured Spheres
Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
12
votes
2answers
305 views
Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
3
votes
1answer
62 views
Intersecting geodesics on a surface from non-intersecting geodesics
Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?
1
vote
0answers
70 views
delta hyperbolicity of a product of poincare balls? [closed]
I know how to compute the $\delta$-hyperbolicity of a $2D$ poincaré disk of radius 1, but I was wondering how to generalize such computations to:
Higher dimensions
Poincaré disk with radius $r$ and ...
3
votes
1answer
144 views
Cusps in hyperbolic manifolds and fundamental group
I am reading the book "The Arithmetic of Hyperbolic Three Manifolds" by Maclachlan and Reid and I am having some problems in understanding something about cusps.
The definition they give of a cusp is ...
2
votes
1answer
69 views
Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$?
I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me.
The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ ...
1
vote
0answers
34 views
First eigenvalue for domains in hyperbolic space
I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...
3
votes
0answers
72 views
Density of closed orbits on hyperbolic surfaces
It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense.
My questions:
If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...
23
votes
4answers
785 views
Immersions of the hyperbolic plane
Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was ...
3
votes
1answer
184 views
Question on a proof of density of periodic orbits
In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...
0
votes
0answers
44 views
Characteristics of Poincare's ball model (of hyperbolic spaces)
Are there any references (papers, books, etc.) that have ready-calculated equations for geometric quantities of the hyperbolic space in terms of the coordinates of Poincare's ball models?
By ...
7
votes
4answers
259 views
Lattices of PU(n,1) with large abelianization
I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...
0
votes
0answers
113 views
Expected distance in hyperbolic space
In a hyperbolic space, $r$ and $\theta$ can represent a point in a polar coordinate system. If we suppose $\theta_1\sim \operatorname{Uniform}(t_1,t_2)$, $\theta_2\sim \operatorname{Uniform}(t_3,t_4)$,...
3
votes
2answers
171 views
Is the development map in Hyperbolic geometry related to development in Cartan geometry?
I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ ...
8
votes
0answers
83 views
Cluster algebra and Fenchel Nielsen coordinates
Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
5
votes
1answer
174 views
How many simple closed geodesics in a given primitive homology class?
It is well-known that an essential closed curve on a hyperbolic surface (possibly with boundary) is homotopic to a unique closed geodesic. Moreover, if the curve under consideration is simple, then so ...
4
votes
1answer
115 views
Reference request: geometric finiteness of Fuchsian groups
My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong):
Proposition. The convex core of a complete hyperbolic surface ...
37
votes
1answer
840 views
Four circles on the sphere
Consider configurations consisting of 4 distinct circles on the sphere.
Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to enumerate/...
5
votes
2answers
350 views
$S^3 \setminus S^1$ doesn't have hyperbolic structure
I need to prove that $M = S^3 \setminus S^1$ doesn't admit any metric of constantly negative sectional curvature s.t. $M$ is complete respect for this metric. I know that it is consequence of famous ...
5
votes
1answer
196 views
What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$?
I have been trying to learn about congruence groups. Here is an example:
\begin{eqnarray*} \Gamma\big(1+2i\big) &=& \text{SL}_2\big(\mathbb{Z}[i]\big)(1+2i) \\ \\
&=& \left\{ \left(...
1
vote
1answer
49 views
Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map
Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
5
votes
1answer
198 views
Conformal boundary and cusp of figure-8 complement
As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...
5
votes
1answer
150 views
Intersection of $\pi_1$-injective surfaces
Let $M$ be a closed non-Haken hyperbolic $3$-manifold. Are there two, nonhomotopic, $\pi_1$-injective closed surfaces in $M$, and which do not intersect?
3
votes
0answers
204 views
The uniqueness of Poincaré metric
The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance.
...
3
votes
0answers
103 views
Harmonic analysis on constant curvature hyperbolic spaces of arbitrary dimension
I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the ...
2
votes
0answers
75 views
Lifts of geodesics on surfaces onto the universal cover [closed]
self-intersecting geodesic on hyperbolic surface of genus 2
Given a self intersecting geodesic on a hyperbolic surface of genus 2 as in the picture, how can I understand precisely what the lift to ...
3
votes
1answer
156 views
Immersed incompressible surfaces in surface bundles
Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not ...
0
votes
1answer
88 views
Hyperbolic structures on infinite type surfaces
Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do ...
9
votes
1answer
221 views
Are two triangles with equal corresponding medians, congruent?
Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
Two triangles with equal corresponding medians are congruent.
More precisely: Assume that $\Delta ABC$ and $ ...
9
votes
1answer
318 views
Parabolic subgroups of relatively hyperbolic and CAT(0) groups
Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...
2
votes
0answers
85 views
Explanation of an unexplored note of Gauss on hyperbolic volume
My question refers to Gauss's note "Cubirung Der Tetraeder", which can found at p. 228 in the section on the foundations of geometry in volume 8 of his collected works. In this note, Gauss wrote down ...
1
vote
0answers
40 views
Real section of moduli space of Riemann surfaces
In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset ...
1
vote
0answers
41 views
Annuli and pinched annuli vs circles and horocycles
Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed.
If we want to parametrise annuli with punctures at one boundary, give the punctures ...
1
vote
1answer
95 views
Parabolic elements of the Poincare' disk automorphism group as limit of elliptic ones
The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point.
...
5
votes
1answer
108 views
Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
2
votes
1answer
138 views
Build a Fuchsian group starting from punctures on a disk
Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$.
$\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the ...
1
vote
0answers
90 views
Cutting a circle from the hyperbolic plane
Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface ...
3
votes
1answer
183 views
What is a half cusp in hyperbolic geometry?
I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...
3
votes
1answer
242 views
closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface
Could you please recommend me some references for proofs of this fact: "closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface". Thanks in advance!
10
votes
2answers
651 views
Does Helly's theorem hold in the hyperbolic plane?
The Helly theorem in the Euclidean plane asserts that if $S_1, \dots, S_n$ are $n \ge 3$ convex subsets such that $S_i \cap S_j \cap S_k \ne \emptyset$ for all distinct triples $i,j,k$, then the total ...
2
votes
0answers
56 views
A boundary for integrals of eigenfunctions over geodesics?
Let $X$ be a compact hyperbolic surface, and $\gamma$ a closed geodesic on it.
Consider the integral
$$\int_\gamma f(x)\, dl(x)$$
where $f$ is a (normalized) Laplace eigenfunction on $X$. ...
5
votes
2answers
230 views
Smallest tile to *isohedrally* tessellate the hyperbolic plane
Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
3
votes
0answers
65 views
Does the orbital function divided by the volume of a ball decrease?
Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...
17
votes
2answers
1k views
Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
6
votes
0answers
175 views
Rational stable translation length
Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...
-3
votes
1answer
126 views
Lemma involving sinh and cosh [closed]
I need a small lemma for a paper I am working on and I claim:
Lemma 1: For every $C >0$ there exist a $c_0 >0$ such that for all $a,b,c,d > c_0$:
$$\sinh(a)\cosh(b) \geq \sinh(c)\cosh(d)$$
...
