Tagged Questions

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

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Freeness of the group generated by two hyperbolic isometries

If $f$ and $g$ are two hyperbolic isometries of the hyperbolic space $\mathcal H^n$, we know that $f$ has 2 fixed points on $\partial \mathcal H^n$ and similiarly for $g$. Is it true that $f$ and $g$ ...
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Proof verification: Cross Ratio

Prove: If $[z_1,z_2,z_3,z_4] \in \mathbb{R} \cup \{\infty \}$, then $z_1,z_2,z_3,z_4$ are either concyclic or collinear. My proof below uses the geometric interpretation of cross ratio. I am not ...
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homeomorphisms of the real line

Given a homeomorphism $h$ of the extended real line. Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$? Any hints are ...
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What are the hyperbolic rotation matrices in 3 and 4 dimensions?

So the hyperbola-preserving transformation in 2 dimensional space is given by the matrix \begin{pmatrix} \cosh(\phi) & \sinh(\phi) \\ \sinh(\phi) & \cosh(\phi) \end{pmatrix} I'm wondering ...
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Fundamental group of the unit tangent bundle on the genus 2 torus?

I'm interested in the 3-dimensional model geometries; specifically $\widetilde{SL}(2,\mathbb{R})$. I'm looking for a good (see, easily visualizable) example of a compact manifold formed as a quotient ...
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Intuition Behind the Hyperbolic Sine and Hyperbolic Cosine Functions

After enough time studying mathematics, we develop an instinct for the sine and cosine functions and their relationship to our standard Euclidean Geometry. I have come across the functions $\sinh(x)$ ...
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Does 3 points on Poincaré Disk geodesic lie on the same Poincaré Half Plane geodesic?

This may be a trivial question, IMO the answer should be yes. Given a geodesic $\delta$ on the Poincaré Disk's model with $A, B, C \in \delta$ And given that $f(x)$ is an isometry from the Poincaré ...
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Equidistant curves in the Half-Plane model.

Definition: An equidistant curve can be one of the three following: A hyperbolic circle, a horocycle or an equidistant line. In the Half-Plane model, a hyperbolic circle is represented by an euclidian ...
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Definition of hyperbolic lenght.

Theorem 1: Let $\text{arc(AB)}$ be an arc of an equidistant curve (Which can be a circle, a horocircle or an equidistant line) and $(A^{n})$ a sequence of partitions of the arc $\text{arc(AB)}$ such ...
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Area of surface of revolution between planes/concentric cylinders of Sphere/Pseudosphere

If area of surface of revolution with maximum radius $R,$ between two concentric cylinders radii $a,b$ is $$2 \pi R (a-b), \tag 1$$ then find equation of its meridian. EDIT2: i.e., find r(z) ...
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In hyperbolic geometry, exactly how big is a dodecahedron composed entirely of right angles? [closed]

Specifically, I need to know the distance from the center to the vertices, and the distance to the faces.
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Geometry - Inversion/Cross Ratios

Problem 5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The ...
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Geodesic sphere in $\mathbb H^2$

I saw the definition of a geodesic sphere, and I think I'm not able to "see" how do they look like. For example, it's obvious that in $\mathbb R^n$ geodesic spheres are simply normal spheres, and that ...
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curvature of planar hyperbola

If a planar hyperbola is parametrized as follows $$x(t) = a\sec(t), \quad y(t) = b\tan(t), \quad a, b \text{ constants}$$ what is the curvature of the hyperbola curve?
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Hyperbolic isometries and finite order elements

I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of $\text{Isom } H^n$, the finite order elements ...
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Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius ...
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On the connection between Bloch's space semi-norm and Bergman's hyperbolic metric.

On the proof of the following theorem $f\in \mathcal B \Leftrightarrow \beta(f)=\sup\left\lbrace\dfrac{|f(z)-f(w)|}{d_{\mathbb D}(z,w)}:z,w\in \mathbb D, z\neq w\right\rbrace$, where $\mathcal B$ is ...
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Calculate the midpoint of the line given in the Beltrami-Klein model

I'm given that the distance between two points in the Beltrami-Klein model is $$d(XY)=\frac{1}{2}ln\Big(\frac{\overline{XQ}\cdot\overline{YP}}{\overline{XP}\cdot \overline{YQ}}\Big)$$ where $P$ and $Q$...
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Triangle on Beltrami pseudosphere with angle sum $180^\circ$

What characteristic lines on the pseudosphere can form a triangle whose internal angle sum is $180^\circ$?
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Are Fuchsian groups without elliptic and parabolic elements at most countable? [duplicate]

Let $G \subset PSL(2, \Bbb R)$ be a discrete subgroup without elliptic or parabolic elements. Does it follow that it is at most countable? Subgroups as above have the property that the quotients of ...
Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
I've been studying the hyperbolic plane and the action of the group $PSL(2,\mathbb{R})$ on it. I found that the modular group $PSL(2,\mathbb{Z})$ is a discrete subgroup of $PSL(2,\mathbb{R})$ so it's ...