  The well-known fact that $S^1$, $S^3$ and $S^7$ are parallelizable manifolds
admitting flat connections is revisited. The role of torsion in the
construction of those flat connections is made explicit, and the possibilities
allowed by different metric signatures are examined. A necessary condition for
parallelizability in an open region is that the torsion tensor must be
covariantly constant. This property can be used to obtain a relation between a
torsion-free and flat connections. Our treatment covers Riemannian and
pseudo-Riemannian (non-Euclidean signature) hyperbolic manifolds of dimensions
three and seven. Apart from the spherical cases mentioned above, the explicit
flat $\mathfrak{so}(p,q)$ connections with $p+q=3,7$ are constructed for the
coset manifolds $SO(p,q+1)/SO(p,q)$ or $SO(p+1,q)/SO(p,q)$.
