Order of the standard isometric minimal immersions of cross as helical geodesic immersions
type: Item
instance of: scholarly article
title: Order of the standard isometric minimal immersions of cross as helical geodesic immersions
zbMATH Open document ID: 0535.53047
author: Katsuya Mashimo
published in: Tsukuba Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 3850039
zbMATH Keywords: helical geodesic immersion
zbMATH Keywords: minimal immersions
zbMATH Keywords: rank one symmetric spaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496159824
OpenAlex ID: W1582760560
DOI: 10.21099/TKBJM/1496159824
description: scientific article; zbMATH DE number 3850039