Symmetry and minimality properties for generalized ruled submanifolds (Q1891512)

Let \(\overline {M}\) be a standard space of constant curvature and \(R\) a generalized ruled submanifold of \(\overline {M}\), i.e., on \(R\) there is defined a foliation \(F\) whose leaves, called the rulings of \(R\), are totally geodesic submanifolds of the ambient space \(\overline{M}\). The authors prove that \(R\) is a minimal submanifold, if \(R\) is symmetric in the following sense: For each \(p \in R\) the reflection in the ruling \(F_p\) locally maps \(R\) into itself, mapping every ruling into a ruling. The essential tool in the proof is the following fact: For each vector \(v\) perpendicular to \(F\) there exists a unique curve \(\gamma\) orthogonal to \(F\) with \(\dot \gamma (0) = v\) such that the covariant derivative of \(\dot \gamma\) is tangential to \(F\) everywhere. Such curves are mapped into itself by the reflection in \(F_{\gamma(t)}\) for every \(t\).