# 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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### Hyprbolic geodesic attempt and maximum distance

Problem: Let $z,z',w,w'$ be points in $\mathbb{H}^2$. Let $w\in [z,z']$. Then $d(w,w')\leq max \{ d(w,z),d(w',z) \}$ Where $d= d_{\mathbb{H}^2}$ is defined in terms of the cayley map between poincare ...
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### Properties of pseudospherical circles and geodesics

I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) and I have some questions about pseudospherical circles at page 10: the author says that Beltrami ...
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### The Branch Schema of a Subgroup of the Modular Group

Let $\Gamma$ be a subgroup of the modular group $PSL(2, \mathbb{Z})$. What is the best and easiest way to grasp the notion of the Branch Schema of the subgroup $\Gamma$. Why do we have only four cases ...
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### Why do isometries of Beltrami-Klein model correspond to projective transformations fixing the circle at infinity?

I am reading the note "Beltrami's Models of Non-Euclidean Geometry" (PDF link via unibo.it) by Nicola Arcozzi, and I am not quite understanding what is happening at page 9, in discussing ...
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### The hyperbolic plane $\mathbb{H}^2$ can't be isometrically immersed in $\mathbb{R}^3$

It's easy to note that there is a local isometry between the hyperbolic plane $\mathbb{H}^2$ and the pseudosphere, since they have constant curvature equal to $-1$. Hilbert's Theorem.- There exists no ...
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### Showing that the sum of the angles of a hyperbolic triangle are less than $\pi$

Using the law of cosines for sides: $\cosh(a) = \cosh(b) \, \cosh(c) - \sinh(b) \, \sinh(c) \, \cos(\alpha)$ I have to show that $\alpha + \beta + \gamma < \pi$ Unfortunately I find no ansatz for ...
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Consider a hyperbolic quadrilateral of $abcd$ in the hyperbolic plane $\mathbb{H}^2$ with the metric being the metric defined via the cayley map. Suppose $\angle b$, $\angle c$ ,$\angle d$ are all ...
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### Harmonic $1-$ form on the upper half-plane $\mathbb{H}$

Let $\Gamma$ be a normal subgroup of finite index of the modular group $PSL(2,\mathbb{Z})$. A function $f:\mathbb{H}\rightarrow \mathbb{C}$ is called an entire modular form for the subgroup $\Gamma$ ...
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### Second order differential equation with hyperbolic function

Consider the second order differential equation $\frac{d^2f}{dt^2}+a\frac{df}{dt}+bf=0$, where $a,b\in\mathbb{R}$. For which values of $a,b$ do we have $f(t)=\sinh(At+B)$ being a solution of this DE? ...
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### Some version of MVT(?)

Let $T \in SL(2,C)$ be a normalised Mobius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$. The above is an exercise from Outer Circles by Marden (ex. 1.1). I ...
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### $PSL(2,\mathbb{R})$ and the set of oriented geodesics

How can $G/D$ be identified with the set of (oriented) geodesics?