Curvature locus and principal configurations of submanifolds of Euclidean space

Abstract

We study relations between the properties of the curvature loci of a submanifold $M$ in Euclidean space and the behaviour of the principal configurations of $M$, in particular the existence of umbilic and quasiumbilic fields. We pay special attention to the case of submanifolds with vanishing normal curvature. We also characterize local convexity in terms of the curvature locus position in the normal space.