Helix, shadow boundary and minimal submanifolds

Abstract

Inspired by a Blaschke's work about analytic convex surfaces, we study {\em shadow boundaries} of Riemannian submanifolds $M$, which are defined by a parallel vector field along $M$. Since a shadow boundary is just a closed subset of $M$, first, we will give a condition that guarantee its smoothness. It depends on the second fundamental form of the submanifold. It is natural to search for what kind of properties might have such submanifolds of $M$? Could they be totally geodesic or minimal? Answers to these and related questions are given in this work.