# The product geometry $S^2 \times \mathbb E$

## What is $\mathbb S^2\times\mathbb{E} $

This geometry is the cartesian product of the two-sphere and the real line. It can be also seen as the universal cover of the product $M = S^2 \times S^1$ where $S^2$ is the two-sphere and $S^1$ the unit circle.

Click on the button below to reveal a concrete model of $S^2 \times \mathbb E$.

The two-sphere is the unit sphere in the 3-dimensional euclidean space. It can be described as $$ Y = \left\{ (x,y,z) \in \mathbb R^3 \mid x^2 + y^2 + z^2 = 1 \right\}$$ endowed with the following rieamanian metric $$ ds^2 = dx^2 + dy^2 + dz^2.$$

Consequently a possible model of $S^2 \times \mathbb E$ is the following subset $X$ of $\mathbb R^4$ $$ X = \left\{ (x,y,z,w) \in \mathbb R^3 \mid x^2 + y^2 + z^2 = 1 \right\}$$ endowed with the following rieamanian metric $$ ds^2 = dx^2 + dy^2 + dz^2 + dw^2.$$ The isometry group of $X$ is $O(3) \times {\rm Isom}(\mathbb R)$ where $O(3)$ is the isometry group of $S^2$ 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 S^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 $\mathbb S^2 \times \mathbb E$ can be found in the gallery