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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.

The prototypical example of hyperbolic geometry in two dimensions of Gauss-Lobachevsky-Bolyai in which the parallel postulate of Euclidean geometry is replaced by a new postulate of at least 2 parallel lines through an external point not on the given line with sum of interior angles of a geodesic triangle smaller than $\pi$ radians.

1921 questions
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Which surfaces admit hyperbolic metrics bounded by geodesic arcs?

Consider a compact oriented surface of genus $g$ with $k$ boundary components, ordered from $1$ to $k$. Next remove $n_i\geq 0$ points from the $i$-th boundary component, for $i=1,...,k.$ For which sequences $(g,n_1,...,n_k)$, the above surface…
Adam
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Is the conjugate axis of a hyberbola itself a trivial hyperbola?

The definition of a hyperbola is A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. By this definition, the perpendicular line passing through the centre of the…
orionphy
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Coordinates in {4,3,6} hyperbolic honeycomb

I'd like to make a maze or a game that is played in the Order-6 cubic honeycomb. I like the fact that there are infinitely many cells at each vertex. However, I need some kind of coordinates here. The coordinate system must be capable of these…
Hume2
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Minimal element in geometric finite fuchsian group

If $\Gamma\subseteq PSL(2,\Bbb R)$ is a geometric finite (i.e. finitely generated; i.e. $\Gamma\backslash\mathbb{H}$ has finite volume) Fuchsian group which is not co-compact and has $\infty$ as a cusp, then why does the…
Nightgap
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Exciting Topics in Hyperbolic Geometry

I am a first year student and a learner of hyperbolic geometry. I was wondering if you could suggest some exciting topics to research about in this field (some people suggested fundamental polygons and areas of hyperbolic triangles). Any other…
user38268
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Ideal tetrahedron maximum volume

I want to that any tetrahedron in $\Bbb{H}^3$ can be transformed into a tetrahedron that has $0, 1, \infty, z$ as vertices. Also I need to show that the above defined tetrahedron has maximum volume if $z=e^{\frac{2\pi i}{6}}$ and I need to find the…
toric_actions
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Hyperbolic metric and its relation to the distance between matrices

There is a bijection $f$ between $PSL(2,\mathbb{R})$ and $T^1\mathbb{H}$, which sends a matrix $M$ to a vector with base point $\frac{M_{11}i+M_{12}}{M_{21}i+M_{22}}$. In particular, the identity matrix $I$ is sent to a vector with base point which…
The way of life
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Proving a version of the Ultraparallel theorem

I have been asked to prove a version of the Ultraparallel theorem. Let $l_1$ and $l_2$ be hyperbolic lines with normals $n_1$ and $n_2$. Show that there exists a unique hyperbolic line $l_3$ orthogonal to $l_1$ and $l_2$ if and only if…
Polymorph
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If from (1, $\alpha$) two tangents are drawn on exactly one branch of the hyperbola $\frac{x^2}{4} -\frac{y^2}{1} = 1$ the alpha belongs to

If from (1, $\alpha$) two tangents are drawn on exactly one branch of the hyperbola $$\frac{x^2}{4} -\frac{y^2}{1} = 1$$ the alpha belongs to As far as I can see two tangents can be drawn to only one branch if the point lies inside the…
Avyansh Katiyar
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Can a finitely generated discrete group $\Gamma\subset I(\mathbb{H}^{n})$ contain infinitely many elliptic elements with common fixed point?

Let $\Gamma\subset I(\mathbb{H}^{n})$ be a finitely generated discrete group of isometries of the hyperbolic $n$-space. Let $\Gamma_{\infty}$ be the stabilizer of $\infty$, and assume it contains only elliptic and parabolic elements. Let $A\subset…
The way of life
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Properties of hyperbolic conics?

In Euclidean geometry, conics have many interesting properties. We can define them as a geometric places (points on an ellipse all have constant sum of distances from two foci, points on a hyperbola all have constant difference of distances, points…
Marek14
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Which surfaces arise as quotients $\mathbb{H}^2/\Gamma$?

Which surfaces arise as quotients $\mathbb{H}^2/\Gamma$ where $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R})$ which acts freely on $\mathbb{H}^2$? The uniformization theorem tells us that any hyperbolic structure on a compact surface $S$…
Joey Brew
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What is the proof of infinite number of parallel lines in hyperbolic geometry?

I know that parallel axiom in Hyperbolic geometry is that there at least two parallel lines to line $a$ through a given point $A$. But as I know it can be proven that there are infinitely many of them, what is the proof ?
Юрій Ярош
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Parametrization of hyperbolic geodesic line on sphere

A curve in $\mathbb R^3$ starts at equator of sphere radius $a$ being inclined at $\alpha $ to longitude goes to North pole along a hyperbolic geodesic. Find its radius $ r(\theta)$ as function of longitude $\theta$ in cylindrical/polar…
Narasimham
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Prove that a loxodromic transformation has an attractor and a repeller as fixed points

I have to Prove that a loxodromic transformation has an attractor and a repeller as fixed points. I have no idea how to start the proof, or what i need to do, formally. Basically, I only know the geometric picture of what an attractor and a…
Trux
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