while the configuration in which $C_1$ and $C_2$ are images of $C_0$ by pure translations (strictly possible only in ${\mathcal{G}_2}$, and a convenient approximation in ${\mathcal{G}_4}$ if $\theta \approx 90^o$, and the distances of $C_1$ and $C_2$ to $C_0$ are almost equal, as is the case in the Roukema–Edge hypothesis) is equivalent to that of a torus.

For space forms of the classes ${\mathcal{G}_2}-{\mathcal{G}_5}$ the following facts are easily derivable from the generators of their corresponding covering groups (see [@GR02] for details):

1.  The vector $a$ is orthogonal to both $b$ and $c$.

2.  \[angle\] The angle between $b$ and $c$ is a free parameter for the class ${\mathcal{G}_2}$, while its value is fixed to be $120^o$, $90^o$ and $60^o$ for the classes ${\mathcal{G}_3}$, ${\mathcal{G}_4}$ and ${\mathcal{G}_5}$ respectively.

3.  Denoting by $|a|$ the length of the vector $a$, and similarly for any other vector, one has that $|b| = |c|$ for the classes ${\mathcal{G}_3}-{\mathcal{G}_5}$, while both lengths are independent free parameters in the class ${\mathcal{G}_2}$. Moreover, in all classes ${\mathcal{G}_2}-{\mathcal{G}_5}$, $|a|$ is an independent free parameter.

4.  Denoting the canonical unitary basis vectors in Euclidean space by $\{\hat{\imath},\hat{\jmath},\hat{k}\}$, one can always write $a = |a| \hat{\imath}$, $b = |b| \hat{\jmath}$ 