A Gauss-like map associated to a surface in \(\mathbb{R}{}^ 3\) (Q811514)

For each surface \(M\subset\mathbb{R}^ 3\), without umbilics, a Gauss-like map is defined that associates to each point the couple of principal directions, considered as an element of the flag manifold \(O(\mathbb{R}^ 3)/(O(\mathbb{R})\times O(\mathbb{R})\times O(\mathbb{R}))\). A necessary and sufficient condition for this map to be harmonic is given. The differential geometry of the flag manifold is studied by embedding it in a convenient way in a Euclidean space.