# Tagged Questions

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

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### Constant of a hyperbola

Hyperbolas are a companion to a circle, sharing many properties when it comes to their trig functions and equation. But, if the circle has $\pi$ as a constant relation, does a hyperbola have some ...
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### How to show that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures?

I am reading the lecture notes. On page 19, let $G=SL_2$. It is said that the metric $\frac{|dz|^2}{y^2}$ and the 2-form $\frac{dz \wedge d\overline{z}}{(-2i)y^2}$ are $SL_2$-invariant measures on the ...
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### Is the fundamental group of a compact Riemann surface *after* removing a finite number of points still a Fuchsian group?

Let $S$ be a compact R.S. admitting a Fuchsian model $\mathbb{H} / \Gamma$. We know that $\pi_1(S) \cong \Gamma$. Let $\mathcal{B} \subseteq S$ be a finite set of points, is $\pi_1(S - \mathcal{B})$ ...
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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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### Geodesic on hyperboloid and Poincare's Disk Model

I have two questions that 1. Why geodesic on hyperboloid corespond the arc in the Poincare's Disk Model? The hyperboloid : $x^2 + y^2 - z^2 = -1, \hspace{.15cm} z>0$ When any plane through ...