Higher spin $m\not=0$ excitations on curved backgrounds and cosmological supergravity

Some time ago, we showed that a weak (linear) massless spin 2 wave could only propagate on a Ricci-flat or Ricci-constant background: it must necessarily be a perturbation of General Relativity. This was just the continuation of the higher spin chain: massless $s=3/2$ also required Ricci-flatness (which is the basis of supergravity) while $s>2$ needs Riemann flatness. This note re-analyzes the problem in a perhaps more physical way: by considering massless and -- only apparently -- massive small excitations and showing how their parameters relate to the cosmological constant. We will thus prove, in a simple physical way, the necessity, as well as the sufficiency, of the known supergravity results.