Helix, shadow boundary and minimal submanifolds

TL;DR

This paper investigates the properties and smoothness conditions of shadow boundaries in Riemannian submanifolds, exploring their potential to be totally geodesic or minimal, inspired by classical convex surface theory.

Contribution

It provides conditions for the smoothness of shadow boundaries and characterizes when they can be totally geodesic or minimal submanifolds.

Findings

Established a condition for shadow boundary smoothness based on second fundamental form

Identified criteria for shadow boundaries to be totally geodesic

Analyzed when shadow boundaries can be 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.