# Tagged Questions

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### Definition of PU(2,1)?

I know what the unitary group of complex matrices $U(n)$ is, and what $PU(n) = PSU(n) = SU(n)/(\mathbb{Z}/n)$ is. However, I found in an article mentioned $PU(2,1)$, the group of bi-holomorphisms of ...
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### Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
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### Please explain the shortest path between two points in non-euclidean geometry. [closed]

Please explain it for those with inferior knowledge of mathematics (using easy to understand words): e.g., kids and adults with no knowledge of mathematics (calculus, algebra , etc.) , or rather from ...
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### Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
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### $2$-dimensional Hyperbolic space with fundamental group ${\bf Z}$ and constant curvature $-1$

$$d\rho^2 + \cosh^2\rho\ d\theta^2$$ Only one ? Is there any other example ?
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### Covert Euclidean distance to Hyperbolic distance

I am searching for formulae to convert Euclidean distance into hyperbolic distance. The problem that I am confronting is that I want to calculate if a bug travels x units in Euclidean space, how much ...
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### How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
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### projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
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### Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
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### Volume of a Riemannian manifold and its relation to the area

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem): If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...
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### Parabolic elements correspond to punctures

In Mapping Class Group by Farb and Margalit page 22, they say: Let $S$ be a hyperbolic surface. If a non-trivial element of $\pi_1(S)$ is represented by a loop (up to homotopy) around a puncture, ...
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I know that every coset representative $g\in SL(2,\mathbb{C})$ for $SL(2,\mathbb{C})/SU(2)$ can be chosen of the form $$g = \left( \begin{array}{cc} \sqrt{t} & \frac{z}{\sqrt{t}}\\ 0 & ... 3answers 744 views ### Expression of the Hyperbolic Distance in the Upper Half Plane While looking for an expression of the hyperbolic distance in the Upper Half Plane \mathbb{H}=\{z=x +iy \in \mathbb{C}| y>0\}, I came across two different expressions. Both of them in Wikipedia. ... 1answer 259 views ### Thrice-punctured sphere This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45: the thrice-punctured sphere is the largest subdomain of \mathbb{S}^2 supporting a hyperbolic metric. I ... 0answers 202 views ### the hyperbolic plane is complete I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let \mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ... 1answer 238 views ### Questions about Hyperbolic Isometries: The Standard Inversion I have two questions regarding the inversion across the unit circle in the hyperbolic plane. Recall that the hyperbolic plane is a metric space consisting of the open half-plane$$\mathbb{H}^2 = ...
We know that the universal covering of a closed hyperbolic 3-manifold can be identified with the hyperbolic space $\mathbb{H}^3$. Now, what is not very clear to me is how this identification has to be ...
### Backslash notation: $\Gamma {\setminus} \mathbb{H}^n$
I encountered this notation in a paper by Carron: When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ... $\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My ...