  This paper contains some more results on the topology of a nondegenerate
action of $\mathbb{R}^n$ on a compact connected $n$-manifold $M$ when the
action is totally hyperbolic (i.e. its toric degree is zero). We study the
$\mathbb{R}$-action generated by a fixed vector of $\mathbb{R}^n$, that
provides some results on the number of hyperbolic domains and the number of
fixed points of the action. We study with more details the case of the
2-sphere, in particular we investigate some combinatorial properties of the
associated 4-valent graph embedded in $S^2$. We also construct hyperbolic
actions in dimension 3, on the sphere $S^3$ and on the projective space
$\mathbb{RP}^3$.
