# 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 $\mathbb H^2 \times \mathbb E$

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Click on the button below to reveal the fly commands.

The default controls to fly in the scene are the following.
You can choose your keyboard in the *Option controls* pannel in the top right corner of the window

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.