# Tagged Questions

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

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### Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$\zeta(s)=\sum_{n=1}^\infty \frac1{n^s}$$ and can be written in the form $$\zeta(s)=\prod_p\frac1{1-p^{-s}},$$ where the product is over all prime numbers ...
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### This map is an isometry (in the Riemannian sense) of the hyperbolic plane. Why is the following a proof of it?

I'm making my way through a textbook on elementary undergraduate geometry. The author has defined the notion of an isometry between two subsets of $\mathbb{R}^2$ equipped with a Riemannian metric. It ...
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### Examples of Lattices in $\operatorname{Isom}(H^n)$ for all $n \geq 2$?

I just had an exam today where I was asked to give an example of a lattice in $\operatorname{Isom}(H^n)$ for all $n \geq 2$, and with bonus points if I could give cocompact and noncocompact examples. ...
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### How big are regular (hyperbolic) polygons?

Given a hyperbolic surface of constant curvature $K=-1/a^2$ embedded in $\mathbb{R}^3$, is there a known formula for the length of the edges of a regular polygon? I know that the Gauss–Bonnet ...
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In the Poincare disc model, a version of the isoperimetric inequality states that $L(\partial(A))>c\mu(A),$ where $\mu$ is the hyperbolic area and $L(\gamma)=\int_0^1 ... 0answers 58 views ### spin structures on knot complements Let$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 ... 1answer 52 views ### Volumes of hyperbolic manifolds In a talk I attended the speaker said that the volume of a closed hyperbolic manifold is a topological invariant. Are known volumes rational? Irrational? Transcendental? 0answers 43 views ### Hyperbolic Geometry - Parabolic Matrix? In lecture we defined for hyperbolic geometry using the Lorentz model on the upper sheet of the two sheeted hyperboloid: $$Para_x=\begin{bmatrix} 1 + \frac{x^2}{2} & -\frac{x^2}{2} & x\\ ... 0answers 33 views ### What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface? I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If \Gamma is a Fuchsian group, its parabolic elements correspond to the cusps of ... 0answers 51 views ### Prove A \cdot B\leq -1, where A and B are in \mathbb{H}^2 Let A and B be in \mathbb{H}^2. I need to prove that the lorentzian dot product between A and B is less than or equal to -1. I have no idea where to start. 1answer 27 views ### Non euclidean lines (finding endpoints of semicircle) A non euclidean line in \mathbb{RP}^1 in terms of reflections about the unit circle can be written in the form A+B(\overline{w}+w)+C(\overline{w}w)=0 Where w=\frac{1}{\overline{z}} The ... 0answers 23 views ### sympletic form of the hyperboloid I'm working with the Hyperboloid of two leefs H_2 as a coadjoint orbit of \mathfrak{sl}^*(2, R). I know that H_2 is a symplectic manifold by the following theorem: Given a Lie group G and ... 0answers 45 views ### Misinterpretations of Hilbert's Theorem? I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in \Bbb R^3 of constant curvature -k ... 0answers 63 views ### Villarceau circle as a Loxodrome A circular Clifford torus (radius at flat circle = h, section radius a , a<h ) is cut by a plane at an angle \cos \alpha = a/h \tag{1} centrally to the symmetry axis, the line of ... 0answers 37 views ### Example of a doubly degenerate Kleinian group which does not come from a mapping torus Doubly degenerate Kleinian groups are discrete subgroups of PSL(2,\mathbb{C}) whose limit set is all of S^2, the boundary of \mathbb{H}^3. A standard example of such a group is given as ... 1answer 40 views ### Can a hyperbolic surface be isometrically embedded into \mathbb R^4? Can a complete hyperbolic surface be isometrically embedded into flat \mathbb R^4? 1answer 46 views ### When a ray of an horocircle passing through the origin intersects the y axis. In the following figure, h(A,B) is an horocycle centered in A passing over B. \Theta(h) is the angle of parallelism of the segment h and S is the well known intersection of a chord of an ... 1answer 36 views ### Tiling on Poincaré disc [closed] Is there anyone to help me tile on a Poincaré disc? In fact, I'm going to tile triangle tiles on a surface in hyperbolic geometry ; is there any algorithmic method to do so? 2answers 54 views ### Mapping the Poincaré disk to hyperbolic surfaces in \mathbb{R}^3. Take any hyperbolic surface with constant curvature in \mathbb{R}^3, such as Dini's surface, or a hyperboloid of constant curvature. If I understood things correctly, for any such surface, we ... 0answers 13 views ### How to make a triangular pillow with a pre-drilled tube with two truncated tetrahedra? (from Jeff Weeks' paper) In his paper Computation of Hyperbolic Structures in Knot Theory, p.12, Jeff Weeks explains as below how to make a triangular pillow with a pre-drilled tube by gluing two truncated tetrahedra. A ... 1answer 57 views ### Compute of curvature In the answer of this question,for the given metric$$g_M = ds^2 + \frac1M \sinh^2(\sqrt M s) d\Omega^2,$$how to compute the curvature? Whether the hyperbolic space means$M=\{x\in R^n:x_n>0\}$? ... 1answer 32 views ### An$d$-unramified covering of compact Riemann surfaces induce a (monodromy) action on$d$letters. Is the opposite true? Let$S_1, S$be compact connected Riemann surfaces,$f : S_1 \rightarrow S$be a meromorphic function of degree$d$that branch over$B \subset S$. The unmarried covering$f : S_1 \backslash f^{-1}(B) ...
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 ...