# Tagged Questions

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

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### Fundamental domain of a Fuchsian group is non-compact if the group contains parabolic element

Suppose a discrete subgroup $\Gamma$ of $PSL(2,\mathbb{R})$ acts on $\mathbb{H}^2$. Why is the fundamental domain non-compact if $\Gamma$ contains a parabolic element? Thanks in advance.
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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 ...
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### Christoffel symbols for the Poincaré ball model

The metric tensor $g_{ij}$ of the PoincarĂ© ball model is $$g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2}$$ where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian coordinates....
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### spin structures on knot complements

Let $K$ be a knot in $S^3$, and let $M=S^3/N(K)$ be its knot complement, where $N(K)$ is a tubular neighborhood of $K$. $K$ is given for example by a its projection onto the plane. The question is ...
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### Law of Cosines with imaginary arguments?

Does the law of cosines: $c^2 = a^2 + b^2 - 2 a b \cos \theta$ work with imaginary angles? to get something like: $c^2 = a^2 + b^2 - 2 a b \cosh \theta$ Alternatively, is there a hyperbolic ...
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### Is the tangent bundle of hyperbolic space trivial?

Let $H$ be hyperbolic 3-space. Let $TH$ be the tangent bundle of $H$. I have a question: Is $TH$ isometric to H times a flat $k$-space?
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### Difference between Euclidean and Hyperbolic lengths.

What is the difference between Euclidean and Hyperbolic lengths? For instance if i were to measure a curve on with the euclidean distance and alternatively the Hyperbolic distance, What would be the ...
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### What is the metric of the Riemann surface resulting from quotiening the upper half plane by a Fuchsian group?

Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. What is the metric we get on $\mathbb{H} / \Gamma$?
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### Relativistic Projective Geometry

If we assume that space-time has an extra two dimensions so that there is more symmetry between space (with 3) and time (now with 3). What would the corresponding cross ratio equation look like if we ...
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### Covering a $n$-holed torus for $n\geq 2$ with a hyperbolic tesselation?

How can I cover a $n$-holed torus $(n\ge2)$ with $\frac{2-2n}{\frac pq-\frac p{2}+1}$ faces of regular hyperbolic tesselation {p,q}? I don't need the graphics, just the construction. For example, in ...
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### Search for coverings in hyperbolic tessellations

Given a regular or uniform tessellation of hyperbolic plane, is there a way to find a group of cells that will tile the whole plane? For example: in the $(6,6,7)$ tessellation (truncated $\{3,7\}$), ...
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### Scale and the models of the hyperbolic plane

I was reading somewhere (sorry I always forget where) that the scale of the Poincare Half plane is y (the vertical) So at the boundary line the scale is $0$ or $( 1 : \infty )$. at the ...
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### Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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### Find the hyperbolic length of the geodesic segment

I'm reading my textbook and I'm trying to make sense of this example. So the place with the red star shows the actual process of calculating the hyperbolic length. My question is how they get the ...
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### We can get the geodesic only by the first geodesic equation in hyperbolic plane?

The two geodesic equations are: $$\frac d{ds} (Eu' + Fv') = 1/2(E_u {u'}^2 + 2F_u u'v' + G_u {v'}^2)$$ $$\frac d{ds} (Fu' + Gv') = 1/2(Ev{u'}^2 + 2F_v u'v' + G_v{v'}^2)$$ And in the hyperbolic ...
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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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### 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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### Metric relations in lambert quadrilateral

I already found the relations in a rectangle triangle (6 formulas for the sides) and for a general ordinary triangle (sine and cosine hyperbolic laws). But now I'm trying to find them for a triangle ...
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### Studying the hyperboloid model, what is represented by the conic sections?

I am trying to get my head around the hyperboloid model of hyperboloic geometry https://en.wikipedia.org/wiki/Hyperboloid_model (article is much to technical please improve) And was thinking the ...
I'm trying to understand the concept of Dirichlet domains associated to the action of a Fuchsian group $G$ on $\Bbb H$ (the upper half-plane of $\Bbb R^2$ endowed with its usual hyperbolic metric). ...
In Euclidean geometry it is simple: In a triangle $\triangle ABC$ there is a single point $H_a$ on $BC$ such that the triangles $\triangle ABH_a$ and $\triangle ACH_a$ have the same area. the ...