Differential invariants of Maximally symmetric submanifolds

Let G be a Lie group acting smoothly on a manifold M. A closed, nonsingular submanifold S ⊂ M is called maximally symmetric if its symmetry subgroup G_s ⊂ G has the maximal possible dimension, namely dim G _s=dim S., and hence S=G_s · z_o is an orbit of G_s · Maximally symmetric submanifolds are characterized by the property that all their differential invariants are constant. In this paper, we explain how to directly compute the numerical values of the differential invariants of a maximally symmetric submanifold from the infinitesimal generators of its symmetry group. The equivariant method of moving frames is applied to significantly simplify the resulting formulae. The method is illustrated by examples of curves and surfaces in various classical geometries. Mathematics Subject Classification 2000: 22F05, 53A55, 58A20.