Force free projective motions of the sphere

Abstract

Suppose that the sphere S n has initially a homogeneous distribution of mass and let G be the Lie group of orientation preserving projective diffeomorphisms of S n . A projective motion of the sphere, that is, a smooth curve in G , is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of S n and, more generally, examples of subgroups H of G such that a force free motion initially tangent to H remains in H for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H = S O n + 1 ). The main tool is a Riemannian metric on G , which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.