Immersions into spheres with parallel higher fundamental forms (Q1922680)

In this note the author proves that the minimal immersions of a compact rank one symmetric space into a unit \(n\)-sphere with higher fundamental forms isotropic have all higher fundamental forms parallel, so they are 2-symmetric submanifolds of the unit \(n\)-sphere. As a consequence, helical minimal geodesic immersions into spheres and standard minimal immersions into spheres are examples of 2-symmetric submanifolds. The author also gives conditions under which a nicely-curved 2-symmetric submanifold of a space of constant curvature \(c\geq 0\) is a helical immersion and vice versa.