Helix, shadow boundary and minimal submanifolds

Details Helix, shadow boundary and minimal submanifolds Helix, shadow boundary and minimal submanifolds

2007-06-11

arxiv.org/abs/0706.1524v1

Mathematics

Differential Geometry

15 pages

Scientific paper

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.

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