The product geometry $\mathbb H^2 \times \mathbb E$
What is $\mathbb H^2 \times \mathbb E$?
This geometry is the cartesian product of the hyperbolic plane and the real line. It can be also seen as the universal cover of the product $M = \Sigma_g \times S^1$ where $\Sigma_g$ is a compact surface of genus $g \geq 2$ and $S^1$ the unit circle.
Click on the button below to reveal a concrete model of $\mathbb H^2 \times \mathbb E$.
There exist many models for the hyperbolic plane. One of them is the hyperboloid model given by $$ Y = \left\{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 - z^2 = -1,\ z > 0 \right\}$$ endowed with the following rieamanian metric $$ ds^2 = dx^2 + dy^2 - dz^2.$$
Consequently a possible model of $\mathbb H^2 \times \mathbb E$ is the following subset $X$ of $\mathbb R^4$ $$ X = \left\{ (x,y,z,w) \in \mathbb R^3 \mid x^2 + y^2 - z^2 = -1,\ z > 0 \right\}$$ endowed with the following rieamanian metric $$ ds^2 = dx^2 + dy^2 - dz^2 + dw^2.$$ The isometry group of $X$ is $O(2,1) \times {\rm Isom}(\mathbb R)$ where $O(2,1)$ is the isometry group of $\mathbb H^2$ and ${\rm Isom}(\mathbb R) = \mathbb R \rtimes \mathbb Z/ 2 \mathbb Z$ is the isometry group of the real line.
Some views of $\mathbb H^2 \times \mathbb E$
Warning: Some of the real-time simulations below requires a powerful graphic card. If your computer is not fast enough, you can reduce the size of your browser window. Click on the button below to reveal the fly commands.
The default controls to fly in the scene are the following. If you have a different keyboard, the keys should be the ones having the same location as the given ones on a QWERTY keyboard.
| Command | QWERTY keyboard | AZERTY keyboard |
|---|---|---|
| Yaw left | a | q |
| Yaw right | d | d |
| Pitch up | w | z |
| Pitch down | s | s |
| Roll left | q | a |
| Roll right | e | e |
| Move forward | arrow up | arrow up |
| Move backward | arrow down | arrow down |
| Move to the left | arrow left | arrow left |
| Move the the right | arrow right | arrow right |
| Move upwards | ' | ù |
| Move downwards | / | = |
HD pictures of $\mathbb H^2 \times \mathbb E$ can be found in the gallery
Features of $\mathbb H^2 \times \mathbb E$
Some features of $\mathbb{H}^2\times\mathbb{E}$ are described in the following Bridges paper by two members of our team (Henry and Sabetta) together with Vi Hart and Andrea Hawksley.