A note on duality of vector fields and minimal submanifolds (Q2266454)

Dual vector fields on a surface immersed in 3-dimensional Euclidean space have been studied by the present author and \textit{K. Amur} [Tensor, New Ser. 31, 292-294 (1977; Zbl 0389.53001)]. The present paper extends that study on a 2-dimensional submanifold of a Riemannian space. Minimal surfaces are characterized in terms of dual vector fields. The following propositions are proved: (1) Given a smooth vector field \(X\) and a normal section \(\xi\) in \(E\), there always exists a smooth vector field \(Y\) dual to \(X\) with respect to \(\xi\). (2) Let \(E\) be minimal in Riemannian space \(N\). Then any two vector fields on \(E\) are dual to each other with respect to the normal section \(\xi\) if and only if they are orthogonal.