Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [hyperbolic-geometry]

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

0
votes
0answers
7 views

Find a Dirichlet polygon

Let $\Gamma=\{\gamma_n(z)=3^nz\}$ be a Fuchsian group. I want to find a Dirichlet polygon. First I need to choose the center of the Dirichlet domain, I suppose it would be the point $p=0$ cuz it's ...
3
votes
0answers
39 views

Trigonometric integral related to Gieseking's constant

This question at MathOverflow https://mathoverflow.net/questions/302982/how-to-prove-the-identity-l2-frac-cdot3-frac215-sum-limits-k-1-inf conjectures certain relation between fast converging ...
0
votes
1answer
31 views

Constructing bisectors for a triangle in Poincare Disk (wolfram mathematica)

I'm having difficulties constructing angle bisector in Poincare Disc(Hyperbolic geometry), specifically with writing code in Wolfram Mathematica. I've managed to create a triangle, but now have no ...
1
vote
1answer
45 views

Show that the Riemann sphere complement the unit disc is a hyperbolic Riemann surface

I am trying to understand what $\hat{\mathbb{C}} \backslash \bar{\mathbb{D}}$ in the area of Riemann surfaces. $\hat{\mathbb{C}}$ is the Riemann sphere and $\bar {\mathbb{D}}$ is the closure of the ...
0
votes
0answers
13 views

How to compute the Hessian of a function on the Poincare unit disk model of the hyperbolic plane?

I have a function $f$ mapping the open unit disk $\{z\in \mathbb{C}: |z|<1\}$ to $\mathbb{R}$. Really this is a function from the hyperbolic plane to $\mathbb{R}$, represented using the Poincare ...
2
votes
0answers
16 views

Lower bound on the decay of the injectivity radius in hyperbolic manifold

Let $M$ be a complete orientable hyperbolique manifold with finite volume (3 dimensional in my case, typically a knot complement), that is essentially a quotient of $\mathbb{H}^3$. Let $r(x)=d_M(x,...
0
votes
0answers
32 views

Eigenvalues of Laplacian in modular space

Let $\mathbb{H}$ be the upper half plane and $\Delta=-y^2(\partial_x^2+\partial_y^2)$ the (positive) Laplace-Beltrami operator and consider $\Gamma:=\operatorname{PSL}(2,\Bbb Z)$. Moreover, let $F:=\{...
4
votes
0answers
53 views

Complex-analytic argument that $\mathrm{SL}(2,\mathbb R)$ acts isometrically on $\mathbb H^2 \subseteq \mathbb C$

Let $\mathbb H^2 := \left\{ z = x+iy \in \mathbb C : y > 0\right\}$, equipped with the hyperbolic Riemannian metric $\breve{g} = \frac 1{y^2}\left(dx^2 + dy^2\right)$. It is a classical result that ...
0
votes
0answers
14 views

Distance function in an asymptotically hyperbolic space

The distance function in the Poincare disk model of hyperbolic space is given by $$d(p, q) = \cosh\left(1 + 2\frac{\|p - q\|^2}{(1- \|p\|^2)(1-\|q\|^2)}\right)$$ where $p, q$ are two points in the ...
0
votes
0answers
52 views

Proving the distance formula in the Poincaré disk by integrating the line element

The line element on the Poincaré disk is given by $$ds^2 = \frac{4\|dy\|^2}{(1-\|y\|^2)^2}.$$ I don't understand how to go from this line element to the equation $$d(p, q) = \cosh\left(1 + 2\frac{\...
0
votes
0answers
14 views

Book recommendations “Computing Volume of some sets on hyperbolic surface with boundary”

I would like to learn how to compute the critical exponential of Fuchsian group and volume of some subset on the surface and its unit tangent bundle. I found that there is a book related to this, that ...
3
votes
0answers
61 views

Prove transformation $z \mapsto \frac{az+b}{cz+d}, z = x+ iy, ad-bc = 1$ is isometry of Hyperbolic plane.

Prove transformation $$f: z \mapsto \frac{az+b}{cz+d},\ z = x+ iy,\ ad-bc = 1$$ is isometry of Hyperbolic plane $$M=\{(x,y)\in \Bbb R^2:y>0\} \text{ with Riemannian metric } g= \frac{1}{y^2}(dx \...
0
votes
1answer
84 views

What the boundary of the disc in Poincaré disk model exactly is by a geometric point of view?

Poincaré disk model is defined in a open disc, and the boundary of the disc represent something infinitely distant. But what the "something" exactly is? How to topologically or geometrically extend ...
-1
votes
0answers
21 views

Find eccentricity and length of semi transverse axis of given hyperbola

Find eccentricity and length of semi transverse axis of given hyperbola? y=x-1/x or we can write x^2 - xy=1
1
vote
0answers
12 views

Reflect point inside Poincarè Disk Model

I have a question: is there a formula which map points onto the Poincarè Disk Model starting from points which are outside the unit Disk? For example, I have a point p with norm = 2, so it is outside ...
0
votes
0answers
12 views

Orthogonal Projection of Point in Ambient Space onto Hyperboloid

Let the following denote the set of points on the $n$-dimensional hyperboloid manifold $$ \mathbb{H}^n_K=\left\{x\in\mathbb{R}^{n+1}\,\middle|\,\langle x,x\rangle_*=-r^2=\frac{1}{K} \,\land\,x_1>0\...
0
votes
0answers
10 views

How does scale Hyperbolic Distance if the radius change?

I trained a neural network which makes a regression to points in Poincarè Disk Model with radius $r = 1$. I want to optimize using the hyperbolic distance $$ \operatorname{arcosh} \left( 1 + \frac{...
1
vote
0answers
68 views

Minimal element in geometric finite fuchsian group

If $\Gamma\subseteq PSL(2,\Bbb R)$ is a geometric finite (i.e. finitely generated; i.e. $\Gamma\backslash\mathbb{H}$ has finite volume) Fuchsian group which is not co-compact and has $\infty$ as a ...
1
vote
1answer
47 views

Is the Euclidean Pythagorean Theorem true for right triangles in Poincare's half plane?

Suppose we have a right triangle ABC in the Poincare half plane such that angle C = $\frac \pi2$. Is it possible to construct this triangle with the property that $d(A, B)^2 + d(A, C)^2 = d(B, C)^2$ ...
3
votes
0answers
79 views

Formula in hyperbolic quadrilateral with one point on the absolute

I am asked to show that for a hyperbolic quadrilateral with $3$ right angles and one point on the absolute with finite sides of length $a,b$ then $$\sinh(a)\sinh(b)=1$$ I have constructed a line ...
3
votes
1answer
37 views

How to solve Poisson's equation on compact Riemann surfaces of genus greater than one?

$M$ is a compact Riemann surface, $f\in C^{\infty}(M)$. I want to find the solution of $\Delta \varphi=f$. When $M=T^2=\mathbb{R}^2/2\pi \mathbb{Z}^2$, I can use Fourier series on $\mathbb{R}^2$ to ...
3
votes
1answer
80 views

An isometry of the upper half plane which is a “bounded distance from the identity” must be the identity?

In Farb-Margalit's book: https://www.maths.ed.ac.uk/~v1ranick/papers/farbmarg.pdf on page 204, in the proof of Prop 7.7, they say that if $S_g$ is a closed hyperbolic surface of genus $g$, and $\phi$...
1
vote
1answer
79 views

Geometric interpretation of an elliptic point on a Riemann surface / hyperbolic surface

Let $\Gamma$ be a Fuchsian group of signature $[g;m_1,\dots,m_r;s]$. When we quotient $\mathbb{H}^2$ by $\Gamma$ we obtain a genus $g$ surface with $s$ cusps and $r$ elliptic points with orders $m_1,\...
0
votes
1answer
56 views

Locus of points of the same distance from two diverging lines.

I am asked to find the locus of points of the same distance from two diverging lines in the the hyperbolic plane. I am using the Poincare model and am trying to use the unique common perpendicular to ...
0
votes
0answers
15 views

Transitive action of SL2(R) on the hyperbolic lines in Poincare upper half plane [duplicate]

I'm trying to prove that the action of SL2(R) is transitive for the hyperbolic lines in H (Poincaré's Half-Plane Model). Thank you in advance for your answers!
2
votes
0answers
55 views

Does there exist a fuchsian group that is not hyperbolic?

Does there exist a fuchsian group that is not hyperbolic ? By hyperbolic group I mean a group whose Cayley graph is hyperbolic in the sense of Gromov, and a fuchsian group is a discrete subgroup of ...
2
votes
0answers
51 views

Hyperbolic fixed point of ODE

Suppose we have a linear autonomous two dimensional ODE: \begin{equation} \frac{dx}{dt} = Ax \end{equation} for some matrix $A \in \mathbb{R}^{2 \times 2}$. Now we say a system is hyperbolic if the ...
0
votes
1answer
35 views

Buseman function hyperbolic plane

Fix $z_0 \in \mathbb{H}$ and $\alpha \in \mathbb{R} \cup \infty$. Show for $q \in H$ $\text{lim}_{q \rightarrow \xi} (d_\mathbb{H}(q,z) -d_\mathbb{H}(q,z_0))$ exists $\forall z \in \mathbb{H}$. Case: ...
1
vote
1answer
44 views

Defining convex combinations in hyperbolic space

Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space. Given a sequence $u_0,\ldots,u_m$ of points in $\mathbb H^n$ and $t_0,\ldots,t_m$ nonnegative real numbers whose sum is $1$, let us define ...
0
votes
1answer
27 views

what would be a euclidean argument for why the characteristic axiom always holds in hyperbolic geometry?

what would be a euclidean argument for why the characteristic axiom always holds in hyperbolic geometry? Characteristic axiom states Given a line k and a point p not on k, there are at least two ...
1
vote
0answers
37 views

Reference request: Fuchsian Model of compact Riemann surface

Let $S$ be a compact Riemann surface with genus $\geq$ 2. Then by the Uniformization theorem it has universal covering space the upper half-plane $\mathbb{H}$(up to conformal equivalence). Now I read ...
1
vote
2answers
69 views

“Infinite City” - hyperbolic geometry, or something else?

Background: I'm doing worldbuilding for my D&D world, and I want one location to be an "infinite city": finite circumference from the outside, but as you move toward the "center" there's always ...
2
votes
1answer
45 views

Projection of point onto closest point on geodesic in hyperbolic geometry (hyperboloid model)

Lets say we have a point $p\in\mathbb{H}^n$ in hyperbolic space (with curvature 1). And a geodesic, starting at the origin $(1,0,\ldots,0)$ in direction $\mathbf{v}$. What I'd like to do, is to find ...
0
votes
1answer
50 views

Conformal property of transformation [closed]

I want to know if a LFT, $F$, is conformal on the hyperbolic plane $\mathbb H^2$ , that is if we have the curves $\Gamma_1$ and $\Gamma_2$ that intersect at a point $P$ making the an angle $X$, then $...
4
votes
2answers
104 views

Metric Tensor of Hyperboloid Model for Hyperbolic Space with Curvature $K$

Let $\mathbb{H}^n_K$ denote the set of points of the hyperboloid model, which models the hyperbolic space with sectional curvature $K<0$. So $$ \mathbb{H}^n_K=\left\{x\in\mathbb{R}^{n+1} \,\middle|...
1
vote
0answers
53 views

Translating points in hyperbolic geometry

The hyperboloid given by: $$ x^2 + y^2 - z^2 = -1 $$ can be parameterized as: $$ \begin{align} x &= \sinh(r)\ \cos(\theta)\\ y &= \sinh(r)\ \sin(\theta)\\ z &= \cosh(r)\\ \end{align} $$ ...
1
vote
1answer
34 views

Bound on distance of geodesics in metric space

Bridson and Haefliger in their Metric Spaces of Non-Positive Curvature in proof of proposition II.1.4 make use of the following fact: given any positive $\ell<D_\kappa$ there is constant $C$ ...
4
votes
1answer
56 views

Universal cover of a not-necessarily-complete hyperbolic manifold

The Cartan-Hadamard theorem (as typically stated) tells us that the universal cover of a geodesically complete and connected Riemannian manifold $M$ with non-positive sectional curvature is ...
-1
votes
1answer
67 views

Simplification of $\sin(\pi^x)$ , with $x$ being a positive irrational number

How to simplify if $a > 0$ and $\cos(a) < 0$ Was a previous post. Correction, it was suppose to be if a > 0 & cos(a) > 0. An answer was given. https://math.stackexchange.com/a/1274372/...
2
votes
1answer
52 views

non-uniformly hyperbolic definition

I'm working on Uniformly hyperbolic finite-valued $SL(2,R)$ -cocycles( article from Arthur Avila and Jairo Bochi)and at the beginning of my researches,i want to know the exactly meaning of the title ...
0
votes
1answer
28 views

Hyperbolic isometry and line segments

I was trying to apply Poincare's Polygon theorem, for that I had to give a pairing of sides, i.e., to have an isometry of the hyperbolic plane that will take a side of a polygon to another side (of ...
0
votes
1answer
32 views

How to prove the existence of infinite geodesics that do not intersect in hyperbolic space

My given question is: In hyperbolic space, given a geodesic $L$ and a point $p$ not lying on $L$, show there is an infinite number of geodesics through $p$ which do not intersect $L$. The "model" ...
18
votes
2answers
823 views

Magnifying glass in hyperbolic space

My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such ...
0
votes
1answer
21 views

Right triangle circumscribed by a horocycle

Is it possible that a right triangle is circumscribed by a horocycle? Or, is this statement a theorem in the hyperbolic gemetry? For any horocycle $\gamma$, there are no three distinct ordinary ...
1
vote
0answers
28 views

Hyperbolic 3-manifolds of finite volume as link complements

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference). Let $N$ be a hyperbolic 3-manifold of finite volume ...
0
votes
1answer
53 views

Can this equation $b^2$ = $c^2-a^2$ be derived intuitively?

Today while proving the equation of hyperbolas,$$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\tag{1}$$ I came across this expression $$b^2=c^2-a^2\tag 2$$ Though this expression seems much like Pythagorean ...
1
vote
1answer
47 views

Understanding a comment by Thurston

In page 359 (right after Theorem 2.3) of the following paper Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (...
0
votes
0answers
28 views

Is every hyperbolic 3-manifold of finite volume a link complement in some closed 3-manifold?

The question says all I need to know, but I will try to clarify it a little more. Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic ...
2
votes
0answers
29 views

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$?

Why is the fundamental group of a closed hyperbolic $n$-manifold is a (uniform) lattice in $SO(n,1)$? Why is $SO(n,1)$ the (orientation-preserving) isometry group of real hyperbolic $n$-space? Is ...
0
votes
1answer
62 views

Prove that hyperbolic isometry with constant distance is the identity

Let $f$ be an isometry of the Poincaré half-plane model of two-dimensional hyperbolic geometry, denoted by $\mathbb{H}^2$. Prove that if the distance $d(z,f(z))=c$ for some constant $c\geq0$ for all $...