Some Remarks on the Topology of Hyperbolic Actions of $\mathbb{R}^n$ on $n$-manifolds

Abstract

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 $\mathbb{RP}^3$.