The Nil geometry

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.

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