Tagged Questions

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

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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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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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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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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 interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...
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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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Area of polygon of hyperbolic disc

Let us consider hyperbolic disc. I use uniform tessellation {5,4}.Here 5 stands for pentagon, 4 for number of polygons sharing the same vertex. {hyperbolic disc} There exists formula which defines ...
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Distance in Poincaré disk from origin to a point given

Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$ The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the ...
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Element of infinite order for a given group presentation

Let $G=\langle a,b,c,d \mid abcda^{-1}b^{-1}c^{-1}d^{-1}\rangle$ be our presentation. The claim is that the commutator $[a,b]$ has inifinite order in $G$. I think this might be related to small ...
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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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Finitely Many genus-g Quotients of Compact Riemann Surface

I hear there is a semi-famous theorem from my advisor, but he didn't know the name and I was unable to find it online. Does anybody know of the following? Let $S$ be a compact Riemann surface. Then ...
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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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Kobayashi distance on the Siegel upper half space

Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article Intrinsic Metrics on ...
Triangle on Beltrami pseudosphere with angle sum $180^\circ$
What characteristic lines on the pseudosphere can form a triangle whose internal angle sum is $180^\circ$?
I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where ...