in the direction $v$. The action of ${\mathrm{SL}}(2,{\mathbb{C}})$ extends to oriented geodesics via: $$A\cdot {\mathrm{geod}}(p,v) := {\mathrm{geod}}(A\cdot p, A\cdot v).$$

#### Parallel transport.

Let $p,q\in{\mathbb{H}}^3$ and $v\in T_p{\mathbb{H}}^3$. We denote the result of parallel transporting $v$ from $p$ to $q$ along the geodesic of ${\mathbb{H}}^3$ joining $p$ to $q$ by $\Gamma_p^qv \in T_q{\mathbb{H}}^3$. The parallel transport of vectors from the identity matrix is easy to compute with Proposition \[propParallTranspI2\].

\[propParallTranspI2\] For all $p\in{\mathbb{H}}^3$ and $v\in T_{{\mathrm{I}}_2}{\mathbb{H}}^3$, there exists a unique $S\in{\mathcal{H}}_2^{++}\cap{\mathrm{SL}}(2,{\mathbb{C}})$ such that $p=S\cdot {\mathrm{I}}_2$. Moreover, $ \Gamma_{{\mathrm{I}}_2}^p v = S\cdot v$.

The point $p$ is in ${\mathbb{H}}^3$ identified with ${\mathcal{H}}_2^{++}\cap {\mathrm{SL}}(2,{\mathbb{C}})$. Define $S$ as the unique square root of $p$ in ${\mathcal{H}}_2^{++}\cap {\mathrm{SL}}(2,{\mathbb{C}})$. Then $p= S\cdot {\mathrm{I}}_2$. Define for $t\in[0,1]$: $$S(t) := \exp\left(t \log S\right), \qquad \gamma(t) := S(t)\cdot {\mathrm{I}}_2, \qquad v(t) := S(t)\cdot v.$$ Then $v(t)\in T_{\gamma(t)}{\mathbb{H}}^3$ because $${\left\langle v(t), \gamma(t) \right\rangle} = {\left\langle S(t)\cdot {\mathrm{I}}_2, S(t)\cdot v \right\rangle} = {\left\langle {\mathrm{I}}_2, v \right\rangle} = 0$$ and $S\cdot v = v(1)\in T_p{\mathbb{H}}^3$.

Suppose that $S$ is diagonal. Then $$S(t) = \begin{pmatrix}
    e^{\frac{at}{2}} & 0\\
    0 & e^{\frac{-at}{2}}
    \end{pmatrix} \qquad (a\in{\mathbb{R}})$$ and using Equations and , $$\gamma(t) = \begin{pmatrix}
    e^{at} & 0\\
    0 & e^{-at}
    \end{pmatrix} = {\mathrm{geod}}({\mathrm{I}}_2,\sigma_3)(at)$$ is 