# The Nil geometry

## What is Nil?

Nil is a 3-dimensional nilpotent Lie group. It can also be seen as the universal cover of the suspension of a 2-torus by a Dehn twist.

Click on the button below to reveal a concrete model of Nil.

Nil is also the 3-dimensional Heisenberg group, that is the set of matrices of the form $$ \begin{bmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix} $$ However this model does not highlight the symmetries of Nil. Therefore, we use a different (but isomorphic) model. As a set of points, Nil is the usual 3-dimensional space $X = \mathbb R^3$ with coordinates $(x,y,z)$. The riemanian metric on Nil is given by $$ ds^2 = dx^2 + dy^2 + \left(dz - \frac 12(xdy - ydx)\right)^2.$$

Consider to points $p_1 = (x_1, y_1, z_1)$ and $p_2 = (x_2, y_2, z_2)$ in $X$. The group law in this model becomes $$ p_1 \ast p_2 = \left(x_1 + x_2, y_1 + y_2, z_1 + z_2 + \frac 12 (x_1 y_2 - x_2 y_1)\right) $$ The left action of Nil on itself is an action by isometries. Its full isometry group is ${\rm Isom}(X) = {\rm Nil} \rtimes O(2)$. In particular the stabilizer of the origin $[0, 0, 0]$ contains a subgroup isomorphic to $S^1$ which corresponds to the rotations around the $z$-axis.

## Some views of Nil

**Warning:**
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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 Nil can be found in the gallery

## Features of Nil

Some features of Nil are described in the following Bridges paper.