Questions tagged [hyperbolic-geometry]
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885 questions
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Uniqueness of distance realizing geodesic in hyperbolic surface. [duplicate]
Possible Duplicate:
Hyperbolic surfaces
Given a hyperbolic surface S with geodesic boundary. Let a and b be two distinct simple closed geodesic boundaries. Does there exist a unique distance ...
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Definition of k -quasisymmetric maps on S^1
I know the definition of k -quasi-symmetric maps $f$ on $R$,it is
there exists $k>0$ such that $\frac{1}{k}\leq\frac{f(x+t)-f(x)}{f(x)-f(x-t)} \leq k \forall x,t\in R.$
So I just want to ...
- 1,471
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altering curvature on a tessellation representation of a compact surface
I have been reading about tessellation representations of compact surfaces, such as how the square tiling the plane represents the torus. For surfaces of genus > 1 (the ones that interest me), we ...
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Existence of a geodesic on a non-orientable surface
Let $\Sigma$ be a non-orientable surface possibly with boundary or punctures. Is it possible that a one-sided loop in $\Sigma$ is always realized as a geodesic?
In the orientable case, it is well-...
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When inclusion between two Kobayshi hyperbolic manifolds is distance decreasing?
Suppose that $X$ and $Y$ are two Kobayshi hyperbolic complex-analytic manifolds such that $X \subset Y$. It is known $d_Y(x_1, x_2) \leq d_X(x_1, x_2)$ for all $x_1, x_2 \in X$. In other words, the ...
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Are the unit tangent spaces over hyperbolic surfaces always Seifert?
I saw on Wikipedia (in the geometrization conjecture article) that the unitary tangent space over a surface S with finite volume and genus > 1 is a Seifert manifold.
What if we do not assume that S ...
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Determining a convex hyperbolic pentagon by all side lengths and two specified angle sums
We are trying to prove the following statement for convex hyperbolic pentagons which we believe should be true.
Consider a convex hyperbolic pentagon with sides of lengths $a, b, c, d, e$. Suppose the ...
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Theorems that hold in Euclidean and Elliptic Geometry but not in Hyperbolic (and vice versa)
Question: Which theorems (if any) of plane Euclidean geometry continue to hold in Elliptic geometry but don't hold in Hyperbolic? And are there theorems that are valid in Euclidean and Hyperbolic ...
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The upper bound of hyperbolic cosine function in complex plane
I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...
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Is a right triangle with the given cathetus and the opposite angle constructible in the absolute plane?
Question. Is it possible to construct a right triangle with a given cathetus $a$ and a given opposite angle $\alpha$ using only compass and ruler in the absolute geometry (so without the axiom of ...
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Number of tiles inside a region of a hyperbolic tiling
Let $\mathbb{D}$ be the Poincare disc model of hyperbolic geometry with $\{p,q\}$ tiling on it.
In this, paper authors calculated the number of tiles on any circular region centered at a vertex or ...
- 613
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86
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Deformation of hyperbolic structures
Let M be a hyperbolic 3-manifold, let $g_{t}$ be a smooth family of hyperbolic metrics on M. It is known that in this case we can find a family of devlopping maps $dev_{g_{t}}:\tilde{M} \to \mathbb{H}^...
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Hyperbolic random geometric graphs with less clustering
The hyperbolic random geometric graph $G_{\mathbb{H}}$ consists of a $N$ uniformly random points (within a disk of radius $R$ centred at the origin) of the hyperbolic plane $\mathbb{H}$, connected ...
- 640
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Is there any way of explaining the Cayley/Beltrami–Klein metric to undergrads?
How to explain the Cayley-Klein or sometimes called Beltrami–Klein metric concept to find the distance between two points in a hyperbolic space to an audience with no higher education than maybe a ...
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Classification of fundamental domains of a fuchsian group
Let $G$ be the (2,3,7) triangle group. We can see it as symmetry group of (2,3,7) tiling of the hyperbolic plane or symmetry group of $[3^7]$ tiling of the hyperbolic plane. This contains translations,...
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Inverse limit in category of graphs
Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....
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Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$
I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$.
Lattice point means integer coordinates and equation with integer means diophantine ...
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242
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Exponential map and optimization
Apologies in advance for being somewhat vague. I'm trying to get pointers to establish a connection between a common trick used in practice in optimization, and the exponential map in differential ...
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Bounded subsets of $\delta$-hyperbolic metric spaces
I was reading this book by Coorneart, Delzant, and Papadopoulos. I am stuck with this proposition in Chapter 1 (Proposition 1.5)
If $Y$ is a bounded $\delta$-hyperbolic subset of $X$, then $X$ is $\...
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Bound on the distance from points to the boundary of a hyperbolic surface
Fix $\epsilon\in\mathbb{R}_{>0}$, $\Sigma$ a surface with boundary and let $\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$ denote the Teichmüller space of hyperbolic structures of $\Sigma$ with geodesic ...
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161
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Gyromidpoint of a hyperbolic line
In his book Barycentric calculs in Euclidean and hyperbolic geometry, A. A. Ungar defines the gyromidpoint of a segment in a Möbius gyrovector space. The Poincaré disk model in dimension 2 and the ...
- 2,560
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341
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Differential rotations in Chebyshev net
A Chebyshev net obeying Sine-Gordon equation is drawn on a surface of constant negative Gauss curvature $K$ so that the asymptotic differential rhombic element corners lie on lines of maximum/minimum ...
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Hyperbolic Space, Lattice Isometries, and Polyhedral Fundamental Domains
Let $L$ be a lattice of signature $(1,n)$. Suppose I have a (probably infinite index) subgroup $\Gamma\subset O^+(L)$ of the isometries of $L$ which preserve the positive cone $\mathcal{C}^+\subset L\...
- 1,493
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186
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Hyperbolic manifold of dim 3 with finite volume.
The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) ...
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Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?
Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space.
Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...
- 2,436
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177
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Existence of special pants decompositions for non-elementary representations into PSL(2,R)
A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation $\rho\...
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Geometric effects of removing elements of D2n generalizable?
So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so that it is still a ...
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Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces
Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...
- 1,471
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Quick references/sources for the hyperbolic Riemann Surfaces with boundary
Hello,
Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...
- 1,471
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uniform properness of lifts of uniform proper maps
Let $(X,d)$ be a unique geodesic metric space and $Y\subseteq X$ a subset, equipped with the induced path metric $d'$. We say that the inclusion $i\colon Y\hookrightarrow X$ is uniformly proper if $\...
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How to understand this isomorphism? [closed]
A Proposition from a book written by Benson Farb and Dan Margalit
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Equal-area projections of the hyperbolic plane [closed]
I'm aware of the projections of the hyperbolic plane at http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane, but how would I project the hyperbolic plane onto a flat piece ...
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3-dimensional hyperbolic space [closed]
In the 3-dimensional hyperbolic space there are given a plane $\mathcal{P}$ and four distinct lines $a_1, a_2, r_1, r_2$ in such positions that $a_1$ and $a_2$ are perpendicular to $\mathcal{P}$, $r_1$...
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Horospherical distance in CAT($-1$) space
In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then $$d_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d_{\...
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Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \Gamma$, where $\Gamma$ has no parabolic generators?
Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret ...
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