Infinitesimally homogeneous submanifolds of euclidean spaces

Abstract

For a class of submanifolds of ℝN, the infinitesimally homogeneous ones, the second fundamental form and itss-times iterated derivativess≤k+1 at any fixed point determine the immersion uniquely. The integerk>0 will be called the extrinsic Singer invariant. Any infinitesimally homogeneous submanifoldM (which is not necessarily complete) is an open part of a globally homogeneous (complete) submanifold. Indeed, the infinitesimal data at any pointp, determine, canonically, a Lie subgroupG of the isometry group of ℝ N , whose orbit atp is a complete submanifold that extendsM.