\right\rangle} = -\det X$ and ${\mathbb{H}}^3$ is identified with the set ${\mathcal{H}}_2^{++}\cap {\mathrm{SL}}(2,{\mathbb{C}})$ of Hermitian positive definite matrices with determinant $1$. This fact enjoins us to write $${\mathbb{H}}^3 = \left\{ F{F}^* \mid F\in{\mathrm{SL}}(2,{\mathbb{C}}) \right\}.$$ Setting $$\label{eqPauliMatrices}
\sigma_1 = \begin{pmatrix}
0& 1\\
1 & 0
\end{pmatrix}, \qquad \sigma_2 = \begin{pmatrix}
0& i\\
-i & 0
\end{pmatrix}, \qquad \sigma_3 = \begin{pmatrix}
1& 0\\
0 & -1
\end{pmatrix},$$ gives us an orthonormal basis $\left(\sigma_1,\sigma_2,\sigma_3\right)$ of the tangent space $T_{{\mathrm{I}}_2}{\mathbb{H}}^3$ of ${\mathbb{H}}^3$ at the identity matrix. We choose the orientation of ${\mathbb{H}}^3$ induced by this basis.

#### Rigid motions.

In the matrix model of ${\mathbb{H}}^3$, ${\mathrm{SL}}(2,{\mathbb{C}})$ acts as rigid motions: for all $p\in{\mathbb{H}}^3$ and $A\in{\mathrm{SL}}(2,{\mathbb{C}})$, this action is denoted by $$A\cdot p := Ap{A}^*\in{\mathbb{H}}^3.$$ This action extends to tangent spaces: for all $v\in T_p{\mathbb{H}}^3$, $A\cdot v := Av{A}^*\in T_{A\cdot p}{\mathbb{H}}^3$. The DPW method takes advantage of this fact and contructs immersions in ${\mathbb{H}}^3$ with the moving frame method.

#### Geodesics.

Let $p\in{\mathbb{H}}^3$ and $v\in UT_p{\mathbb{H}}^3$. Define the map $$\label{eqExpMapGamma}
    {
    \begin{array}{ccccc}
        {\mathrm{geod}}(p,v) & : & {\mathbb{R}}& \longrightarrow & {\mathbb{H}}^3 \\
        & & t & \longmapsto & p\cosh t + v \sinh t. \\
    \end{array}
}$$ Then ${\mathrm{geod}}(p,v)$ is the unit speed geodesic through $p$ 