Questions tagged [hyperbolic-geometry]

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4 answers
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Reference for shortest educational path to (Riemannian) hyperbolic plane

I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
David Steinberg's user avatar
2 votes
0 answers
43 views

Number of small eigenvalues for flat unitary bundles

It is known that the number of small eigenvalues (eigenvalues less than $1/4$) of the Laplacian on hyperbolic surfaces may be topologically bounded from above. If $X$ is a finite area hyperbolic ...
user447643's user avatar
15 votes
1 answer
350 views

Are hyperbolic $n$-manifolds recursively enumerable?

Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable? Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit ...
Jean Raimbault's user avatar
5 votes
2 answers
406 views

Fenchel-Nielsen length-length coordinates on Teichmueller space?

Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...
JHM's user avatar
  • 2,254
2 votes
1 answer
122 views

Example of maximal multicurve complex

in this paper we have : " On the Teichmüller tower of mapping class groups By Allen Hatcher at Ithaca, Pierre Lochak at Paris and Leila Schneps." Definition. The maximal multicurve complex $...
Usa's user avatar
  • 119
4 votes
1 answer
239 views

Tzitzeica surface

A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface ...
MathDG's user avatar
  • 242
6 votes
0 answers
237 views

Almost parallelizable hyperbolic manifolds

In Sullivan's paper Hyperbolic geometry and homeomorphisms (Proc. Georgia Topology Conf., Athens, Ga., 1977) he makes use of a closed hyperbolic almost parallelizable manifold in every dimension. ($M$ ...
Danny Ruberman's user avatar
14 votes
2 answers
597 views

Are there good mutually interpretable axioms for synthetic Euclidean and hyperbolic geometry?

There are familiar analytic equiconsistency proofs for Euclidean and hyperbolic geometry.  Those proofs are so robustly geometric that it seems like they must have synthetic analogues. Looking into ...
Colin McLarty's user avatar
17 votes
3 answers
2k views

Examples of the Thurston geometries with transitive Lie group action

Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries: (1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup (2) Euclidean: 3 torus $\...
Ian Gershon Teixeira's user avatar
3 votes
2 answers
322 views

Hyperbolic volume of hyperbolic knots

Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ? It seems that there is some necessary conditions: $H_{1}(BG) = \mathbb{Z}$ $H_{2}(BG) ...
GSM's user avatar
  • 333
3 votes
1 answer
260 views

Counterexample to mostow rigidity theorem

I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...
GSM's user avatar
  • 333
1 vote
0 answers
88 views

Is there an effective genus theory for indefinite quadratic forms?

For positive definite quadratic forms, there is a way to check if two forms are isomorphic by arithmetic equivalence over $GL_n(Z)$ by computing configurations of vector up to some norm and then ...
Mathieu Dutour Sikiric's user avatar
7 votes
3 answers
319 views

Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...
Lisa's user avatar
  • 71
5 votes
1 answer
422 views

Isometric immersion of $\mathbb H^2$ into $\mathbb R^\infty$ built by Bieberbach

I'm analyzing the following isometric immersion of $(\mathbb H^2,g_D)$ in $(\ell^2,g_\infty)$ given by $f(x,y)=(x_1,x_2,\dots,x_{2m-1},x_{2m},\dots)$ with \begin{align}\label{5.1} x_{2m-1}=\color{...
Zaragosa's user avatar
  • 123
1 vote
0 answers
107 views

The best lower bound for isometric immersions

I just read Azov's article in the considered two classes of Riemannian metrics, \begin{align*} ds^2&=du_1^2+f(u_1)\sum_{i=2}^ldu_i,&f>0\\ ds^2&=g^2(u_1)\sum_{i=2}^ldu_i^2 ,&g>0\...
Zaragosa's user avatar
  • 123
1 vote
0 answers
58 views

Variation of the geometry of a Dirichlet region as the defining point varies

Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...
user101010's user avatar
  • 5,319
0 votes
1 answer
152 views

Half space vs growing balls in the hyperbolic plane [closed]

Let $p$ be a point in the $2$-dimensional hyperbolic space $H_2$. Consider a normal coordinate system $(x,y)$ at centered at $p$. Let $o_x\in H_2$ be the point of coordinate $(x,0)$ and let $C_x$ be ...
Chevallier's user avatar
1 vote
0 answers
79 views

Maximal geodesically convex function interpolating three points on the hyperbolic plane

Crossposted on MSE: https://math.stackexchange.com/questions/4282998/maximal-geodesically-convex-function-interpolating-three-points-on-the-hyperboli Let $M$ be a two-dimensional Hadamard manifold. ...
ccriscitiello's user avatar
2 votes
0 answers
170 views

Structure of hyperbolic manifolds of finite volume

Let $X$ be a hyperbolic manifold of finite volume. I want to prove that $X$ has ends of the form $N\times \mathbb{R}$ where $N$ has a finite covering by a nilmanifold and $\pi_1N\to \pi_1 X$ is ...
Radeha Longa's user avatar
2 votes
0 answers
60 views

Relation between symmetries of hyperbolic knot and the symmetries of a generic triangulation

For canonical ideal triangulation of a hyperbolic knot, the symmetries of the knot are the same as the symmetries of that triangulation. This is how SnapPy computes the knot symmetry group. Is there ...
Zhengdi Sun's user avatar
4 votes
1 answer
179 views

Elliptic equations in asymptotically hyperbolic manifolds

I am interested in reading about existence and regularity theorems for elliptic equations on manifolds with negative (constant) curvature outside a compact subset. I am aware of some results in this ...
MathqA's user avatar
  • 313
4 votes
0 answers
154 views

Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!

Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...
user101010's user avatar
  • 5,319
4 votes
1 answer
160 views

Ideal triangulation of hyperbolic 3-manifold with generic mapping class group

I am from physics background so I apologize in advance if my question is trivial. Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...
Zhengdi Sun's user avatar
4 votes
1 answer
307 views

A local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$

I found this local isometric immersion from $\mathbb H^{n}$ into $\mathbb R^{2n-1}$, given by Schur (1886) in Über die Deformation der Räume constanten Riemannschen Krümmungsmaasses as follows, $(1\...
Zaragosa's user avatar
  • 123
4 votes
2 answers
275 views

Quadratic cusp shape

Which hyperbolic $3$-manifolds are known to have quadratic cusp shape? Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...
ThiKu's user avatar
  • 10.2k
4 votes
1 answer
223 views

Computation of cusp shape from vertex invariants

Following Takahashi ("On the concrete construction of hyperbolic structure of 3-manifolds"), I was able to construct the Euclidean cusp cross-section for the 5_2 knot complement (please see ...
Ken's user avatar
  • 43
2 votes
0 answers
102 views

Possible values of hyperbolic quadratic forms

$\newcommand\Z{\mathbf{Z}}$Given an integral quadratic form $q$ of signature $(n-1,1)$ and a value $\lambda \in \Z$ is there an algorithm that can determine whether there exist $x\in \Z^n$ such that $...
Mathieu Dutour Sikiric's user avatar
2 votes
1 answer
181 views

Do once-punctured torus bundles have integral traces?

Once-punctured torus bundles are a well-studied class of hyperbolic 3-manifolds, but unfortunately I have been unable to find out whether they always have integral traces (in the sense of [1], Def. 5....
wandersam's user avatar
  • 125
5 votes
2 answers
190 views

Rozendorn's Article

I'm researching the isometric dips of the hyperbolic plane and in particular I'm interested in reading the results of Rozendorn who proved that the hyperbolic plane is isometrically immersed in $\...
Zaragosa's user avatar
  • 123
3 votes
1 answer
173 views

Special kind of 3-manifolds

Is there an open connected orientable 3-manifold $M$ with the following properties: $M$ admits a complete hyperbolic metric with finite hyperbolic volume. $H_{i}(M,\mathbb{Z})=0$ for any $i>0$.
GSM's user avatar
  • 333
5 votes
2 answers
357 views

Vertices of hyperbolic triangle with given angles

This is probably a well-known problem in hyperbolic geometry, but here goes anyway. In the Poincar'e upper-half plane model, I am given three angles $\alpha$, $\beta$, and $\gamma$ with $\alpha+\beta+\...
Henri Cohen's user avatar
  • 11.4k
1 vote
0 answers
256 views

Mirzakhani's work and surfaces with marked points on the boundary

Mirzakhani proved identities for the lengths of geodesic curves on Riemann surfaces of genus $g$ and with $n$ boundary components. She used these to provide an integration scheme over the ...
giulio bullsaver's user avatar
1 vote
1 answer
108 views

Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?

The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at ...
Malkoun's user avatar
  • 4,981
6 votes
1 answer
330 views

Non-compact Dirichlet fundamental domains and free Fuchsian groups

Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model. Assume throughout that $\mathcal{F}$...
JackTodd's user avatar
1 vote
0 answers
154 views

Triangle similarity in Euclidean/ Hyperbolic geometries

A pair of parallel lines (red) is cut by two transversals (blue) through point P in Euclidean and Poincare half plane models. Would alternate angles be equal in either case in similar triangles ...
Narasimham's user avatar
1 vote
2 answers
294 views

The moduli space of finite volume hyperbolic 3-manifolds?

By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$. I will call $$\mathcal{M}...
GSM's user avatar
  • 333
4 votes
1 answer
372 views

Explanation of perpendicularity of a Jacobian vector field

Here are some notes on hyperbolic manifolds. The aim is to prove that if $M_1$ and $M_2$ are simply connected, complete Riemannian manifolds having constant sectional curvature of $-1$, then $M_1$ and ...
Ma Joad's user avatar
  • 1,591
1 vote
0 answers
359 views

Which polygons tessellate the hyperbolic plane?

The packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. It is well known that in Euclidean geometry, all triangles and all ...
Nandakumar R's user avatar
  • 5,401
2 votes
0 answers
71 views

Reference request: Discrete subgroup of $\mathrm{PO}(n,1)$ preserving proper subspace has infinite covolume

I'm looking for a reference for the following claim: $\newcommand{\PP}{\mathrm{PO}(n,1)}$ Let $\PP$ denote the group of isometries of $V = \mathbb{R}^{n,1}$ preserving the upper sheet of the ...
user148575's user avatar
4 votes
0 answers
84 views

Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds

Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary. When $M$...
Ilya Gekhtman's user avatar
11 votes
2 answers
301 views

Birman-Series for variable negative curvature

A famous theorem of Birman and Series says that if $S$ is a compact hyperbolic surface, then the set of points that lie on simple geodesics is nowhere dense and has Hausdorff dimension one; in ...
Andy Putman's user avatar
  • 43.3k
3 votes
0 answers
218 views

Immersion of a part of the hyperbolic plane in $\mathbb{R}^3$

I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert's theorem ensures that there is no isometric immersion ...
Zaragosa's user avatar
  • 123
3 votes
1 answer
134 views

Busemann-Feller lemma in hyperbolic space

The classical Busemann-Feller lemma in Euclidean space says the following. Let $K\subset \mathbb{R}^n$ be a closed convex set. Then for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...
asv's user avatar
  • 21.1k
35 votes
17 answers
2k views

Equivalent definitions of Gromov hyperbolicity

Let $X$ be a metric space. I'd like to collect as many definitions of Gromov hyperbolicity or $\delta$-hyperbolicity of $X$ as possible. I'm happy for the definitions to require some niceness ...
4 votes
1 answer
349 views

Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces. Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
giulio bullsaver's user avatar
2 votes
1 answer
258 views

How to express this hyperbolic extension of the cross-ratio in terms of hyperbolic distances and volumes?

Given $4$ distinct points on the Riemann sphere, thought of as the sphere at infinity of hyperbolic $3$-space $H^3$, one may define the cross-ratio in the usual way. Note that the cross-ratio is the ...
Malkoun's user avatar
  • 4,981
2 votes
1 answer
111 views

Hyperbolization with word-hyperbolic fundamental group

In Davis-Januszkiewica´s paper Hyperbolization of polyhedra it is shown that for every manifold $M$ there exists a map $N \to M$ of non-zero degree such that $N$ is aspherical (plus some more ...
ThorbenK's user avatar
  • 1,175
3 votes
2 answers
353 views

Learning roadmap for Lorentzian geometry

I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF. I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (...
user2022's user avatar
2 votes
0 answers
80 views

Showing an action of a higher rank lattice on hyperbolic space has a fixed point

In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
Mauro's user avatar
  • 191
1 vote
1 answer
209 views

A paradox in hyperbolic geometry? [closed]

I was making some routine calculations when I found the strange following result : in the hyperbolic plane (with the Gauss curvature set at K=-1), let's $C$ a circle of center $O$ and radius $R$ and $...
Dfeldmann's user avatar

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