Force free projective motions of the sphere

LAZARTE, M MARCELA; SALVAI, MARCOS; WILL, ADRIÁN

JOURNAL OF GEOMETRY AND PHYSICS

Elsevier

Año: 2007 vol. 57 p. 2431 - 2436

0393-0440

Suppose that the sphere Sn has initially a homogeneous distribution of mass and let G be the Lie group of orientation preserving projective diffeomorphisms of Sn. 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 Sn 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 = SOn+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.