= Symmetric-matrix model of the two-sheeted hyperboloid
{title2=$B=\begin{pmatrix}t+x&y\\y&t-x\end{pmatrix},\quad\det B=t^2-x^2-y^2$}
The displayed coordinates identify determinant-one real <symmetric matrices> of size two with the <two-sheeted hyperboloid>. The <special linear congruence action on symmetric matrices> is transitive on each sheet, with stabilizer $SO(2)$. In particular the positive sheet is the homogeneous space $SL(2,\mathbb R)/SO(2)$, a model of the <hyperbolic plane>.
Back to article page
