# The universal cover of ${\rm SL}(2,\mathbb R)$

## What is the $\widetilde{\rm SL}(2,\mathbb R)$?

${\rm SL}(2,\mathbb R)$ is the set of all $2 \times 2$-real matrices with determinant one. The space $\widetilde{\rm SL}(2,\mathbb R)$ is its universal cover. There are several ways to think about this space. ${\rm SL}(2,\mathbb R)$ is for instance the unit tangent bundle of the hyperbolic plane $\mathbb H^2$. This point of view gives $\widetilde{\rm SL}(2,\mathbb R)$ a structure of a twisted metric line bundle over $\mathbb H^2$, which can be thought as a hyperbolic analogue of the Hopf fibration.

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## Some views of $\widetilde{\rm SL}(2,\mathbb R)$

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Command | QWERTY keyboard | AZERTY keyboard |
---|---|---|

Yaw left | a | q |

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Pitch down | s | s |

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Move upwards | ' | ù |

Move downwards | / | = |

HD pictures of $\widetilde{\rm SL}(2,\mathbb R)$ can be found in the gallery