On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. II
type: Item
instance of: scholarly article
title: On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. II
zbMATH Open document ID: 0564.53021
author: Toshinobu Nagura
published in: Osaka Journal of Mathematics
publication date: 1982-01-01T00:00:00Z
review text: This is a continuation of our first paper [ibid. 18, 115--145 (1981; Zbl 0482.53046)] concerning the linear mappings \(S_{\sigma}\) from that paper. In section 6 we study subrepresentations of a representation of a compact Lie group \(G\) with respect to its closed subgroup \(K\). We introduce certain constants associated to a representation of \(G\) which describe some intertwining homomorphisms. And under certain conditions, called \((P_ 1)\) and \((P_ 2)\), we prove some properties of these constants, which play important roles for the study of the linear mappings \(S_{\sigma}\). In section 7 we consider an orthogonal symmetric Lie algebra \((\mathfrak g,\sigma)\), and study a 3-dimensional subalgebra of \(\mathfrak g\) as well as its representation induced from that of \(\mathfrak g\). The results in sections 6 and 7 will be used in the later computations. In section 8 we study minimally imbedded symmetric \(R\)-spaces into spheres. It is shown that in these cases the Jacobi differential operator \(\tilde S\) reduces to Casimir operators. In section 9 we recall some basic results on representations of the special orthogonal group \(\mathrm{SO}(n+1)\), and study in detail certain representations of the group \(\mathrm{SO}(n+1)\). It is shown that the properties \((P_ 1)\) and \((P_ 2)\) are satisfied in the cases where immersed manifolds are spheres.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 58J99
zbMATH DE Number: 3899689
zbMATH Keywords: minimal submanifolds
zbMATH Keywords: representation of a compact Lie group
zbMATH Keywords: orthogonal symmetric Lie algebra
zbMATH Keywords: symmetric R-spaces
zbMATH Keywords: Jacobi differential operator
zbMATH Keywords: Casimir operators
zbMATH Keywords: special orthogonal group
MaRDI profile type: Publication
Recommended article: On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. I
Recommended article: On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. III
Recommended article: Q4778457
Recommended article: On degrees of minimal immersions of symmetric spaces into spheres
Recommended article: Q3357019
Recommended article: Q4867767
Recommended article: ISOPARAMETRIC GEODESIC SPHERES AND A CONJECTURE OF OSSERMAN CONCERNING THE JACOBI OPERATOR
Recommended article: On the Sobolev space of isometric immersions
Recommended article: Q4030671
Recommended article: Q4841646
description: scientific article; zbMATH DE number 3899689