On the orbit structure of ℝⁿ onn-manifolds

Abstract

We begin by proving that a locally freeC ²-action of ℝ^n-1 onT ⁿ⁻¹×[0,1] tangent to the boundary and without compact orbits in the interior has all non-compact orbits of the same topological type. Then, we consider the setA ²(ℝⁿ,N) ofC ²-actions of ℝⁿ on a closed connected orientable real analyticn-manifoldN. We define a subsetA _n ⊂A ^r (ℝⁿ,N) and prove that if φ∈A _n has aT ^n-1 x ℝ-orbit, then everyn-dimensional orbit is also aT ^n-1 x ℝ-orbit. The subsetA _n , is big enough to contain all real analytic, actions that have at least onen-dimensional orbit. We also obtain information on the topology ofN.