Some isometric minimal immersions of the three-dimensional sphere into spheres
type: Item
instance of: scholarly article
title: Some isometric minimal immersions of the three-dimensional sphere into spheres
zbMATH Open document ID: 0819.53027
author: Yosio Mutō
published in: Tokyo Journal of Mathematics
publication date: 1995-08-24T00:00:00Z
review text: This paper is an extension of the preceding study due to the author [cf. ibid. 7, 337-358 (1984; Zbl 0562.53044)]. Let \(\psi (\xi, \eta, \zeta)\) be a harmonic homogeneous polynomial of degree \(s= 2\sigma\geq 4\) in three variables \(\xi\), \(\eta\), \(\zeta\). Then the bi-symmetric tensor \(C\) of bi-degree \((s,s)\) satisfying \[ \psi(\langle J_ 1 w,v \rangle, \langle J_ 2 w,v \rangle, \langle J_ 3 w,v \rangle)= C(v, \cdots, v; w,\cdots, w) \] identically belongs to the linear space \(W(3,s)\) of isometric minimal immersions of 3-sphere into spheres. The purpose of this paper is to study such tensors \(C\) and some related topics. The author finds the condition for \(\psi (\xi, \eta, \zeta)\) to be related to an element of \(W(3,s)\) in terms of a mapping \(J^ \#\) of \(V(2, s)\) into \(W(3, s)\), where \(V(2, s)\) is the linear space of harmonic homogeneous polynomials of degree \(s\) in three variables and \(s\) is even. Moreover, another mapping \(J^ \#\) having almost the same property as \(I^ \#\) is defined and some their properties are studied. Finally, the author considers the elevation of elements of \(J^ \# V(2, s)\) and studies some properties of geodesics of isometric minimal immersions associated with \(J^ \# V(2, s)\) or \(I^ \# V(2, s)\).
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 737429
zbMATH Keywords: three-sphere
zbMATH Keywords: harmonic homogeneous polynomial
zbMATH Keywords: minimal immersions
zbMATH Keywords: geodesics
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270127951
OpenAlex ID: W2169411563
DOI: 10.3836/TJM/1270127951
Recommended article: Q3675132
Recommended article: Q3790257
Recommended article: Geodesics in minimal immersions of \(S^ 3\) into \(S^{24}\)
Recommended article: Q5749922
Recommended article: Extension of minimal immersions of spheres into spheres
Recommended article: Isotropic minimal immersions of spheres into spheres
Recommended article: \(2\)-harmonic isometric immersions of \(3\)-dimensional submanifolds with parallel mean curvature vector
Recommended article: Orbits of SU(2)-representations and minimal isometric immersions
Recommended article: On the moduli of isotropic and helical minimal immersions between spheres
Recommended article: Complete minimal hypersurfaces in the hyperbolic space ℍ⁴ with vanishing Gauss-Kronecker curvature
description: scientific article; zbMATH DE number 737429