Questions tagged [hyperbolic-geometry]
Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.
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What is attracting fixed point of a hyperbolic element?
Let $X$ be a hyperbolic metric space.
Let $\Gamma$ be a finitely generated group. We say $\Gamma$ is hyperbolic if it acts properly and cocompactly on a hyperbolic metric space $X$. We define the $\...
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What are the fixed points of the action of $M$?
Here is the question I am trying to solve:
Consider the subgroup $\Omega$ of $PSL(2, \mathbb R)$ generated by the matrix $$\begin{pmatrix}
3 & 0\\
0 & 3
\end{pmatrix}$$
So, $\Omega = \{ M^n | ...
- 3,127
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The representation has finite kernel.
Let $\rho: \Gamma \rightarrow \text{Isom} (\mathbb H^n)$ is a representation of finitely generated group $\Gamma$. Let $x_0 \in \mathbb H^n$ and the orbit map $\tau_\rho: \Gamma \rightarrow \...
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Action of Isotropy group on hyperbolic spaces
Let $\mathbb R^{n,1}$ be the space $\mathbb R^{n+1}$ endowed with the bilinear form $$\langle x, y \rangle_{n,1} = \sum_{i = 1}^{n} x_iy_i -x_{n+1}y_{n+1}$$ We define the hyperboloid model as $$\...
- 59
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Show that $A(C)$ is a vertical line or half circle orthogonal to $\mathbb R.$
Here is the question I am trying to tackle:
Let $C \subset \mathcal{H}^2$ be a half circle orthogonal to $\mathbb R$ or a vertical line and let $A: \mathcal{H}^2 \to \mathcal{H}^2$ be a Möbius ...
- 3,127
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Given two points on the Poincaré disk model, what is the formula for a hyperbolic line passing through the two points?
I'm trying to code a Scratch project where you can place two (or more) points in a Poincaré disk to create hyperbolic lines. So far, the engine is going pretty well, but the actual line in between the ...
- 11
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How to define $\mathbb{H}^2$-rotation about any point?
In the book Geometry of Surfaces by Stillwell, he defines the $\mathbb{H}^2$-rotation about $i$ in view of the conformal disc $\mathbb{D}^2$. Specifically, let
$$ J(z):=\frac{iz+1}{z+i}$$
be the ...
- 357
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Convex cocompact representation of finitely generated groups
Let $\mathbb{H}^n$ be the hyperboloid model for hyperbolic space and $\text{Isom}(\mathbb H^n) = PO(n,1)$.
Let $\rho: \Gamma \rightarrow PO(n,1)$ be a representation of finitely generated group $\...
- 59
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what is the curvature of a hyper-cycle if it lies at the distance of r from its central straight line?
There is a post about the curvature on the hyperbolic plane with several methods to derive such formulas. They arrive at curvature = 1/tanh(r) for a circle with radius r on a hyperbolic plane with ...
- 11
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Isometry group of hyperbolic space
Let $\mathbb R^{n,1}$ be the space $\mathbb R^{n+1}$ endowed with the bilinear form $$\langle x, y \rangle_{n,1} = \sum_{i = 1}^{n} x_iy_i -x_{n+1}y_{n+1}$$ We define the hyperboloid model as $$\...
- 59
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How to find the hyperbolic line segment that is perpendicular to two hyperbolic lines in space?
A hyperbolic line is uniquely determined by its two ideal points (at each end of the line), which can be any two distinct points on the sphere at infinity. Let there be given two hyperbolic lines, $...
- 5,375
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what is $\gamma'(t)$?
Here is the question I am trying to solve:
$(a)$ Given $A \in PSL(2, \mathbb R),$ show that its derivative $D_z A$ satisfies $$g_{Az} (D_z Av, D_z Aw) = g_z(v,w)$$ for all $z \in \mathbb H^2$ and $u,w ...
- 2,087
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Don't understand the definition of linked at infinity
I am readind the book of Farb-Margalit "A Primer on Mapping Class Groups" (but I think it is a general definition) and I have a question on the definition of "Linked at infinity". ...
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Source on Hilbert's theorem about the embedding of hyperbolic space
Is there a book covering (in detail) Hilbert's theorem on the impossibility of an isometric immersion of the hyperbolic plane in $\mathbb R³$ and also Nash's theorem which states that hyperbolic space ...
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Isometry group of quadric model of anti-de Sitter space
I am learning Lorentzian geometry on my own. Consider the space $\mathbb R^{p+2}$ endowed with the bilinear form $$\langle x, y \rangle_{p,2} = \sum_{i = 1}^{p+2} x_iy_i -x_{p+1}y_{p+1}-x_{p+2}y_{p+2}$...
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