Symmetric orbits of orthogonal Veronese actions and their second order envelopes (Q1895238)

A submanifold of Euclidean space that is invariant under reflection at each of its normal spaces is called extrinsic symmetric. These are characterized by the fact that the second fundamental form is parallel: \(\nabla h = 0\). They are necessarily orbits of certain orthogonal Lie group actions [the reviewer, Math. Ann. 247, 81-93 (1980; Zbl 0446.53041)]. Conversely, the orbits of many, but not of all orthogonal Lie group actions share this property. In the present paper, the author continues his program to determine which actions have orbits that satisfy the weaker condition of semi-parallelity \(R \circ h = 0\).