# Tagged Questions

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

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### List of connected Lie subgroups of $\mathrm{SL}(2,\mathbb{C})$.

I am not familiar with the theory of Lie groups, so I am having a hard time finding all the connected closed real Lie subgroups of $\mathrm{SL}(2, \mathbb{C})$ up to conjugation. One can find the ...
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### What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius “A” & “B”?

What is the parametric and cartesian equation of a hyperbolic paraboloid formed by the intersection of two cylinders of radius "A" & "B", which intersect at a distance of "H" from its Axis at an ...
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### How to construct hyperbolically equidistant points on a line?

In Stillwells' "Sources of Hyperbolic Geometry " page 66 figure 3.3 shows an ((incomplete?) construction of hyperbolically equidistant points on a line. I tried to reconstruct the figure but did ...
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### Can an ordinary point be a fixed point?

Is it possible that a point on the unit circle which is an ordinary point (that is, a point which is not a limit point of any set of the form $\Gamma z$ for $|z| <1$) for a Fuchsian group $\Gamma$ (...
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### What is the Area formed when a line is traced between two 3D curves?

This question is quite related to intersection of cylinders, Hyperbolic paraboloid and modelling. I am welding a trunnion to a pipe (both are hollow cylinders in different geometry). They intersect ...
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### Distance between points in hyperbolic disk models

I was puzzeling with the distance between points in hyperbolic geometry and found that the same formula is used for calculating the length in the Poincare disk model as for the Beltrami-Klein model ...
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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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### The shape of Pringles potato chip

Why the shape of Pringles potato chip is hyperbolic paraboloid? I found several articles that say the shape is hyperbolic paraboloid, but cannot find out why it is so. Does anyone have reasonable (...
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### What does the logarithm of a hyperbolic line look like?

At a fixed point $p$ of hyperbolic $n$-space $H$, there is the exponential map from flat $n$-space to $H$ taking straight lines through the origin of the flat space to hyperbolic lines through $p$ in ...
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### Is the usual topology on the upper half plane same as that induced by Riemannian metric?

The upper half plane is a Riemannian manifold, with the Riemannian metric given by $(ds)^2 = (dx^2+dy^2)/y^2$ and thus has a metric topology induced by this metric. Is this topology same as the ...
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### Distance from a point to a line in the hyperbolic plane

I have two questions: What is the distance from a point to a line in the hyperbolic plane? Fix a line $L$ in the hyperbolic plane. What does the set of points of distance $d$ from $L$ look like?
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### Fundamental Domain for Congruence (mod 2) Group

How can I show that the area between the circles $|z|=1$, $|z+\frac{1}{2}|=\frac{1}{2}$, $|z-\frac{1}{2}|=\frac{1}{2}$ in the upper-half plane (here's a picture) is a fundamental domain for the ...
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### Verifying Möbius Transformations using Hyperbolic Geometry

Verify that every transformation from $$H = \left\{Tz = e^{i\theta} \frac{z-z_0}{1-z_0 z} \right\}$$ can be written as $Tz = \frac{az-b}{\bar{b}z+\bar{a}}$ with $|a|^2 - |b|^2 = 1$. The book gives ...
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### Is the cross ratio the unique invariant under projective transformations up to multiples?

I have been studying the actions of $PSL_2(\mathbb{R})$ on the hyperbolic plane recently, and the hyperbolic distance $d(z_1, z_2)$ is the absolute value of the log of absolute value of the cross ...
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### working in hyperbolic geometry

I wonder if anyone can provide me with a simple step-by-step proof in hyperbolic geometry of a fact that does not hold in Euclidean geometry. I imagine an answer to be a series of statements, such ...
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### Non-Euclidean Translations and Rotations

If $f(z) = z + 1$ and $g(z) = -\frac{1}{z}$ show that $$g f g^{-1}(z) = \frac{z}{1-z}.$$ I don't know how to solve this question please help.
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### What is the group $\Gamma$ such that $\mathbb{H}/\Gamma$ is a genus-n torus

We know that the universal cover of genus-n torus is a unit disk ($n\ge2$), which is conformal to upper half plane $\mathbb{H}$, with automorphism group $SL(2,\mathbb{R})$. Thus the genus-n torus can ...
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### Shortest words in a group with finite presentation

Suppose we're given a group with presentation G=, where both the generating set and the relations are finite. Given a word $w$ in the elements of $X$, I would like to know whether this word is ...
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### Broken geodesics in the hyperbolic plane and bending angles

Let $\gamma$ be an infinite broken geodesic in the hyperbolic plane, that is a curve formed by consecutive geodesic segments. Assume also that each of these segments is longer than a certain positive ...
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### Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?

I read the answers to this very interesting question and saw that we can in fact embed the Euclidean plane into hyperbolic 3-space using what is called a horosphere. However, as Hilbert showed us, the ...
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### What is the relationship between hyperbolic geometry and Einstein's special relativity?

I am a third year math student writing a term paper on hyperbolic geometry and I would like to understand its relationship with special relativity. I have read that the hyperboloid model of hyperbolic ...
### What would be called as the shape of $xy=10$ in 3-dimensional space?
As title says, what would be called as the shape of $xy=10$ in 3-dimensional space? It doesn't seem to be paraboloid nor hyperboloid...