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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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Hyperbolic Triangles and maximizing

Let $A$,$B$ and $C$ be the nodes of a hyperbolic triangle $\triangle$ in $\mathbb{H}^2$. Suppose the angle $\alpha$ at $A$ has angle at least $\frac{\pi}{2}$. Show that side $BC$ has maximal length. My attempt: Let us maximise the angle $\alpha$ to…
user643073
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Tangent space of the hyperbolic space

Let ${\mathbb{H}}^n$ the $n$-dimensional hyperbolic space, $$ {\mathbb{H}}^n = \{(x_0 , x_1 , \ldots , x_n) \in {\mathbb{R}}^{n + 1} : - x_0^2 + \sum_{i = 1}^n x_i^2 = - 1 , x_0 > 0\}. $$ Then we have a bilinear and simetric form in ${\mathbb{H}}^n$…
joseabp91
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How to prove that a hyperbolic geodesic triangle induces a tesselation

Let $m_1,m_2,m_3$ be integers greather than $2$ such that $\frac{1}{m_1}+\frac{1}{m_2}+\frac{1}{m_3}<1$. Consider a hyperbolic triangle $\tau$ with internal angles given by $\frac{\pi}{m_1},\frac{\pi}{m_2}$ and $\frac{\pi}{m_3}$ on each respectively…
Carlos Adrián
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How does one construct an octagon and side pairing transformations on that octagon, so that the surface is a closed surface of genus 2?

I am aiming to determine what are the initial characteristics of a side-pairing transformations for a hyperbolic octagon so that a Fuchsian group may be generated and a surface of genus 2 can be formed from the quotient of the hyperbolic plane and a…
Hades Pluto
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Is there any homeomorphism between fundamental domains?

It is known that for the same fuchsean group if I take two different points I will get two different fundamental domains. Is there any homeomorphism between these fundamental domains?
mathplayer
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Models for hyperbolic plane with general curvature

So I'm currently studying the hyperbolic plane with curvature $\kappa$. I'm mainly working with the hyperboloid model, which is defined as $\mathbb{H}(\kappa) = \{(x,y,z) \in \mathbb{R}^3 ~|~ z > 0\}$ with metric $ds^2 = dx^2+dy^2-dz^2$. Now I'm…
Mee98
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Rectangles do not exist in hyperbolic geometry

Here it says that rectangles do not exist in hyperbolic geometry because if a line $l$ and a point $P$ not on $l$ are given, then there are more than one lines that passes through $P$ and parallel to $l$. I know that the rectangles do not exist due…
Levent
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Explicit formula for the curvature of line passing through two given points in Poincare disk model.

The title of this question might seem not so clear, as straight lines are by definition curves of zero curvature; however, i refer to lines in Poincare unit disk model, in which straight lines are modeled as portions of circles orthogonal to the…
user2554
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circle reflections in hyperbolic geometry

Determine the equation of the circle reflection of the circle $x^2 + y^2 = 1$ if the circle of reflection is $x^2 + y^2 + 2x = 0$. I'm learning about circle inversion but I still don't get what this question is saying? Is it saying find an equation…
user60887
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Can you connect nodes in a digital network in the pattern of a hyperbolic tessellation?

Looking here (source: djoyce at mathcs.clarku.edu) If at each crossover point you imagine a machine in the cloud executing code, and the machines are connected to the 4 other nearby machines, can you physically distribute these machines such that…
Lance
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Can we use 4d hemisphere for transformation between 3d hyperbolic models

In hyperbolic geometry 3d hemisphere is used for transformation between 2d disc models of Klein and Poincare. Can we use the 4d hemisphere for transformation between 3d Poincare and Klein models
Alparslan Özafşar
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Seeking a formal definition of a complex hyperbolic space

I'm trying to find a formal definition of a complex hyperbolic space that is $CH^n$. Can anyone help me please? I tried looking online already but no formal direct definition.
MOULI Kawther
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Bending in hyperbolic geometry

Bending of a surface conserves parallelism between any two initially parallel geodesic lines. Can some examples be given from hyperbolic geometry ? How are bent surface hyperbolic geodesic parallel lines understood in the three models?
Narasimham
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Every non-elementary subgroup of $PSL (2, \mathbb{R})$ has infinitely many hyperbolic elements in wich one fixed point is distinct

I want to prove that every non-elementary subgroup (call it $\Gamma$ ) of $PSL (2, \mathbb{R})$ has infinitely many hyperbolic elements in wich one fixed point is distinct. (Should I put the definition of non-elementary in this question?) . I…
rowcol
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How can multiple geodesics meet at the boundary on a hyperbolic plane?

I thought on a hyperbolic plane all geodesics are lines/arcs across the plane whose endpoints are perpendicular to the boundary of the plane. I've heard that all geodesics on a hyperbolic plane either intersect ("meet") once (somewhere in the…
minseong
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