Order of the standard isometric minimal immersions of cross as helical geodesic immersions
The author studies the order of a special class of immersions. The present paper is the third one on this subject [cf. ibid. 4, 133-145 (1980; Zbl 0462.53031) and 5, 291-297 (1981; Zbl 0495.53054)]. Let \(\Phi\) :\(M\to \tilde M\) be an isometric immersion of a Riemannian manifold M into another Riemannian manifold \(\tilde M\). Let \(\kappa\) be an arbitrary geodesic in M parametrized by arc-length. If the curve \(\Phi {\mathbb{O}}\gamma\) in \(\tilde M\) is of rank d and has constant curvatures \(k_ 1,...,k_{d-1}\) which are independent of the choice of the geodesic \(\gamma\), then \(\Phi\) :\(M\to \tilde M\) is called a helical geodesic immersion of order d [cf. \textit{K. Sakamoto}, Math. Ann. 261, 63- 80 (1982; Zbl 0476.53033)]. The author proves that the standard isometric minimal immersions of compact rank one symmetric spaces into spheres are examples of helical geodesic immersions and calculates their order. It follows from earlier theorems that the order of this class of immersions coincides with their degree. Similar results are given by \textit{K. Tsukada} in the paper reviewed below.
scientific article; zbMATH DE number 3850039