Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.
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0answers
128 views
Hyperbolic geometry
Post Number: 45
Posted on Friday, 22 March, 2013 - 04:48 pm:
I was asked the following question and i do not have any clue on these. Could anyone help me in the beginning of this?
Show that there ...
3
votes
2answers
263 views
Does anyone know a good hyperbolic geometry software program?
We are currently using this program called NonEuclid but it is a little frustrating to use sometimes and I was wondering if anyone knows another program for hyperbolic geometry.
3
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6answers
199 views
Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?
I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...
2
votes
2answers
54 views
Riemann surface arising as a quotient of the upper half-plane.
Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$
Denote $\Gamma$ the ...
2
votes
1answer
127 views
Riemann surface with punctures corresponds to a hyperbolic surface with cusps
I am reading a paper on Riemann surfaces and the author used the fact that
$\{$Riemann surfaces with
genus $g$ and $n$ punctures$\}$
is in one-to-one correspondence with
$\{$ hyperbolic surfaces ...
1
vote
1answer
224 views
Hyperbolic area and $SL_2$
Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that:
a. The measure $\mu$ is invariant under all $g \in ...
3
votes
0answers
41 views
Question regarding the projective models of the anti-de-Sitter spaces and good online references for learning them from the scratch? (Specifics below)
As my title says above, I am trying to find answers to and also good online reference where I can find complete description of projective models of hyperbolic space, de-Sitter space and anti-de-Sitter ...
3
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1answer
251 views
Models of hyperbolic geometry
Wikipedia states the following:
[The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
2
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3answers
217 views
Gromov boundary — TFAE
I am a newcomer to hyperbolic geometry and was trying to understand some of it in the context of dynamics, for reading certain literature.
Let a discrete subgroup $G$ of $SL_2(\mathbb R)$ act on the ...
1
vote
1answer
37 views
Isometries of a hyperbolic quadratic form
I am reading an article that says "The group of isometries (of a hyperbolic space) of a hyperbolic quadratic form in two variables is isomorphic to the semi-direct product $\mathbb{R} \rtimes ...
1
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1answer
49 views
How does my Beltrami-Klein model look?
http://imageshack.us/photo/my-images/109/hyperbolicquestion.png/
Did I sketch the picture right based off of the specific instructions given in the problem?
1
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1answer
148 views
A book to study about hyperbolic plane, hyperbolic translations, etc.
In this paper, page $6$, the authors state the following:
The translations of the hyperbolic plane are defined as products of
two central symmetries; the set of hyperbolic translations forms a
...
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0answers
40 views
Fréchet mean of the hyperbolic shape space
The Fréchet mean of a general subspace is defined as
$$F(x)=\int_Mdist(x,y)^2d\mu(y),$$
where $\mu$ is the probability measure on a general metric space $(M,dist)$.
I understand that the Fréchet mean ...
-1
votes
1answer
95 views
Length of a curve on $S^2$
$1.$ Could any one tell me what is the shortest distance between $2$ points on $S^2$?
$2.$ Could any one tell me how to measure explicitly a length of a curve on the $S^2$ using polar co-ordinates?
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