  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.
