Some Isometric Mimimal Immersions of the Three-Dimensional Sphere into Spheres

Abstract

In the present paper we extend the study in [3]. 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 \[ \psi(\langle J_{1}w,v\rangle,\langle J_{2}w,v\rangle,\langle J_{3}w,v\rangle)=C(v,\ldots,v;w,\ldots,w) \] identically belongs to the linear space $W(3,s)$ of isometric minimal immersions of the three-sphere into spheres. The purpose of the present paper is to study such tensors $C$ and to state some related topics.