0
votes
1answer
30 views

What is the hyperbolic plane equivalent to translation in euclidean space

in euclidean plane one can move polygons like rectangles, triangles etc. around by isometries, e.g. translations. For instance if we consider a rectangle with midpoint $0\in\mathbb{R}²$ then the image ...
2
votes
1answer
24 views

Question about hyperbolic space-forms

I have a map between hyperbolic space-forms $\varphi:B^n/\Gamma \longrightarrow B^n/H$ (where $\Gamma, H$ are discrete groups of isometries that act freely), and a lift to a map $\tilde{\varphi}:B^n ...
1
vote
0answers
48 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
1
vote
1answer
89 views

Defining a Hyperbolic Metric in a General Surface S

I hope someone can help me or give a ref. I'm trying to understand the general way in which one defines a hyperbolic metric on a given surface $\Sigma$ ( and, if not too complicated, on a manifold of ...
5
votes
0answers
117 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper ...
1
vote
0answers
80 views

Geodesic flow on a manifold with negative curvature is ergodic

This is a question asking for references. I'm not sure if it's appropriate for StackExchange. If it's not, please tell me, thanks! :) I'm reading about the Mostow's rigidity theorem, and the proof ...
-1
votes
2answers
126 views

Please explain the shortest path between two points in non-euclidean geometry. [closed]

Please explain it for those with inferior knowledge of mathematics (using easy to understand words): e.g., kids and adults with no knowledge of mathematics (calculus, algebra , etc.) , or rather from ...
1
vote
1answer
161 views

Geodesic hyperbolic metric

For a hyperbolic metric on the upper half plane $H = \{(u,v)\in \mathbb{R}^2 \ | \ v>0\},$ how can I prove that the vertical lines are geodesics and that the intersection of any circle centered on ...
1
vote
1answer
66 views

area form of the Poincare half plane

For the upper half plane $\{(u,v)|v>0\}$, its area form is $du\wedge dv/v^2$. How to compute the area between the u axis and the curve $\alpha(t)=(r\cos t, r\sin t)$, $0< t < \pi$? Is this ...
1
vote
1answer
103 views

Distance between a point and the origin in Poincaré Disk

How to calculate the distance between the origin and the point $p=(a,0)$, $a>0$, using the metric $g = \frac{4}{(1-x^2-y^2)^2}(dx^2+dy^2)$? I don't how to use correctly these $dx^2$'s
2
votes
2answers
94 views

Dirichlet Domain of a Fuchsian Group

Recall that a Fuchsian group is a discrete group $\Gamma \leq PSL(2, \mathbb{R})$ and that a Dirichlet domain for $\Gamma$ is a set $D \subset \mathbb{H}^2$ of the form $$\{z \in \mathbb{H}^2: d(z, p) ...
3
votes
1answer
72 views

complete metric on a Riemann Surface

I'm reading a book and found a sentence I don't understand: Every Riemann surface $S$ supports an essentially unique complete metric of constant curvature $1$, $0$ or $-1$. Every point of ...
2
votes
1answer
197 views

How to convert between the hyperboloid model and the Poincare patch for $\mathbb{H}_n$?

This question is motivated by the results in this paper, http://calvino.polito.it/~camporesi/JMP94.pdf In this paper some of its most important results about the asymptotics of symmetric traceless ...
0
votes
1answer
73 views

What is complex conjugation in Hyperbolic Geometry

Suppose we are working in the hyperbolic plane, that is the set $$ \mathbb{H}^2 = \{ u + iv \colon v > 0\} \quad \text{with metric} \quad \frac{du^2 + dv^2}{v^2}\,. $$ Now, formally, I can rewrite ...
2
votes
1answer
52 views

Simple closed geodesic around two hyperbolic cusps.

Consider a connected hyperbolic $2$-manifold $M$ with cusps. Consider a simple closed geodesic in $M$, which dissects $M$ into two components. Assume that one of the components contains exactly two ...
1
vote
1answer
77 views

projection onto the nullspace of the Laplacian on a conformally compact surface

Let $(M,g)$ be a conformally compact surface. An example situation is a hyperbolic surface of infinite area like the quotient $\Gamma\backslash \mathbb{H}$, where $\mathbb{H}$ is the hyperbolic plane ...
3
votes
0answers
77 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 ...
2
votes
1answer
190 views

Arc length parameter s

Consider the metric $$ds^2 = \frac{dx^2+dy^2}{y^2}.$$ Assume $R>0, a\in\mathbb{R}$. Consider the curve $$\gamma(\theta)=(a+R\sin\theta,R\cos\theta)$$ for $-\frac{\pi}{2}\leq\theta\leq ...
1
vote
1answer
102 views

geometrically finite hyperbolic surface of infinite volume

I am starting to read some papers involving analysis on hyperbolic manifolds. In these the notion of a "geometrically finite hyperbolic surface of infinite volume" is mentioned frequently and I am ...
1
vote
1answer
81 views

Projecting external points to a circle: Distance order preserving?

Given a circle, and a set of points $A$ that lie external to the circle; I perform the following simple operation: I compute the point of intersection of the i) circle and the ii) line joining each ...
7
votes
2answers
226 views

Simple non-closed geodesic.

In torus there exists simple non-closed geodesic. One example is take the irrational slope curves in $\mathbb{R}^2$ and project it down to torus. Is this thing can happen in closed hyperbolic surface ...
5
votes
1answer
89 views

Embedding manifolds of constant curvature in manifolds of other curvatures

I know that there is no complete surface embedded in $\mathbb{R}^3$ of constant curvature -$k$ for any $k$. But you can clearly embed the hyperbolic plane (curvature -1) into hyperbolic 3-space ...
1
vote
1answer
88 views

Compact surfaces without conjugate points

I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here. I'm trying to ...
-1
votes
1answer
140 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? ...
3
votes
1answer
164 views

Understand the Hyperbolic space

I've been trying to find the expression for the metric of the hyperbolic n-space, $\mathbb H^n$. For $n=2$ I've found (e.g. here) that $$ds^2=\frac{dx^2+dy^2}{y^2}.$$ But for $n>2$ I can't seem to ...
8
votes
1answer
118 views

Hyperbolic diameter of Amsler's surface

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
4
votes
0answers
363 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric ...
5
votes
1answer
508 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
1
vote
2answers
160 views

Perpendicular in conformal disk model

Firstly, please note that the related question can also be found at mathoverflow. The question is stated as following: In Euclidean Geometry, we know that from a given point there is an unique line ...
5
votes
1answer
641 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
0
votes
0answers
90 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
1
vote
1answer
259 views

Geodesic on half-plane determined by tangent vector

The upper-half plane $\mathbb H$ carries a hyperbolic metric and the geodesics are semicircles with base on the real line. We consider oriented geodesics. Let $x \in \mathbb H$ and let $v$ be a unit ...