# Tag Info

7

From the comments you included, I can see you are comparing the two geometries' synthetic axioms to see if one is a special case of the other. Of course, that is doomed to fail because the two lists contain mutually exclusive axioms about parallels (as you noticed.) The real idea is that hyperbolic, affine and Euclidean geometries can be modeled as subsets ...

4

The so called Klein model for hyperbolic space gives a possible answer. Consider an ellipse in the projective plane, or equivalently a quadratic form on the 3-dimensional vector space with signature (+,+,-). The interior of the ellipse is a model for a hyperbolic plane where the lines are the intersections of usual projective lines with the ellipse. The ...

4

Yes; a hyperbolic surface minus a finite set of points is still a hyperbolic surface. If $S$ has genus $g$ then the fundamental group of $S$ minus a point, as an abstract group, is free on $2g$ generators (the same $2g$ generators generating $\pi_1(S)$, but no longer constrained by their usual relation), and every point removed after that increases the ...

2

To ask whether $\pi_1(S - \mathcal{B})$ is Fuchsian, you first have to think of it as a subgroup of $\operatorname{Aut} \mathbb{H}$, the automorphism group of $\mathbb{H}$ as a complex manifold. I assume you want to do this by noting that the universal cover of $S - \mathcal{B}$ is isomorphic to $\mathbb{H}$, so the action of the fundamental group on the ...

2

The surface $S-B$ has a complete hyperbolic metric of finite area. One can see by a pretty direct construction which generalizes the construction of a hyperbolic metric on $S$ itself. Alternatively, since your question is about a given Riemann Surface structure, one can use the uniformization theorem; since removing the points of $B$ creates "removable ...

2

I've created such tessellations in my Ph.D. thesis. I'd start by identifying the corners of one central triangle, using the hypberolic law of cosines. Then I'd describe the reflections in the edges of said triangle. To make sure that I'd create each triangle of the tiling exactly once, I'd create a finite state automaton representing the Coxeter group of the ...

2

For a positive integer $n$, consider the annuli $$A_k = \{z : r(k-1)/n< |z| < rk/n\}, \quad k=1,\dots,n$$ Any curve from $0$ to $r$ contains disjoint subcurves connecting the boundary components of $A_k$, for $k=1,\dots,n$. The length of such subcurve is at least $$\frac{r}{n}\frac{1}{1-(r(k-1)/n)^2}$$ where the second factor is the infimum of the ...

2

We could just as well consider the lower halfplane instead of upper. Or even both, but considering both halves amounts to considering two copies of the hyperbolic plane, existing completely independently from each other. The presence of $y$ in the denominator of the metric density makes the metric blow up as a point approaches the $x$-axis. Moreover, it ...

2

Yes, you can always do this. Let $G\subset \pi_1(S\setminus B)$ be the stabilizer of a point in the monodromy action; then let $f:T\to S\setminus B$ be the covering space of $S\setminus B$ with fundamental group $G$. Then $T$ will have all the necessary properties to be $S_1\setminus f^{-1}(B)$; the only question is whether $T$ can be compactified to a ...

1

Also for an hyperbola there is a link with the number $\pi$, but it can be see only using complex numbers. The coordinates of a point of a circle of radius $1$ satisfy the equation $x^2+y^2=1$ and from this we define the trigonometric functions such that $\cos^2 \theta +\sin^2 \theta=1$. From a unit hyperbola of equation $x^2-y^2=1$ we can define ( in a ...

1

The Dini surface is just a "rolled up" Tractricoid The Tractricoid is just an hyperbolic figure where the "pointy end at infinity " is an ideal point and the "meridians" are hyperbolic parallel lines (converging to the "pointy end at infinity ", left in your diagram). (imagine only one half of the Tractricoid and cut open along an meridian. The parallels ...

1

$\newcommand{\Reals}{\mathbf{R}}$If $S$ is a surface in $\Reals^{3}$ of constant curvature $-1$, then sufficiently small pieces of $S$ are indeed locally isometric to small pieces of the hyperbolic plane, and the universal cover of $S$ is globally isometric to a proper open subset of the hyperbolic plane. (There's no local isometry from the entire hyperbolic ...

1

Every element of $SL_2(\mathbb{R})$ is the product of elementary matrices $$\begin{pmatrix}0&-1\\1&0\end{pmatrix}, \quad\text{and}\quad \begin{pmatrix}1&a\\0&1\end{pmatrix}$$ so it suffices to check invariance under these. In terms of complex maps $(az+b)/(cz+d)$, these are $z\mapsto -1/z$ and $z\mapsto z+a$. The invariance under translation ...

1

I have been thinking about your problem (which i think is more difficult than you should be required to solve). I will solve it using the Poincare Half plane model ( https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model ) If your level is as high as it should be to solve this problem then you should also be able to transform it to the model of ...

1

First, the hyperbolic manifold is the manifold $(\mathbb R^n , g)$ given by one chart $\mathbb R^n$, where in spherical coordinates $(\theta^0= s, \theta^1, \cdots, \theta^{n-1})$, the metric is given by $$\tag{1} g = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2.$$ Note that this is fairly explicit. The manifold you have in mind \{ x\in \mathbb R^n : x_n ...

1

You might know the geometric reformulation of the Hurwitz theorem, which says that there exists a unique compact, connected hyperbolic 2-orbifold $P_{237}$ of minimum area, namely the $(2,3,7)$ triangle reflection orbifold. The connection between the version you state and the geometric version is that the quotient space $S / \text{Aut}(S)$ is a compact, ...

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