The space \(W_ 2\) of isometric minimal immersions of the three- dimensional sphere into spheres
This paper is a continuation of the author's previous paper [Kodai Math. J. 6, 308-332 (1983; Zbl 0532.53043)] and treats the case of isometric immersions \(S^ 3\to S^ M\), but emphasis is layed on the case if M is 24. A remarkable property of SO(4) is that its Lie algebra is spanned by \(J_ 1,J_ 2,J_ 3,I_ 1,I_ 2,I_ 3\) and any element of SO(4) is a linear combination of \(1,J_ 1,J_ 2,J_ 3,I_ 1,I_ 2,I_ 3\). Here each of \((1,J_ 1,J_ 2,J_ 3)\) and \((1,I_ 1,I_ 2,I_ 3)\) is a quaternion structure and \(J_{\lambda}I_{\mu}=I_{\mu}J_{\lambda}\). It is proved that any point in \(W_ 2\) of \textit{P. M. do Carmo} and \textit{N. R. Wallach} [Ann. Math., II. Ser. 93, 43-62 (1971; Zbl 0218.53069)] in the case \(M=24\) can be given explicitly with the use of this structure and homogeneous harmonic polynomials of \(R^ 3\), of degree 4. In this paper it is also proved that dim \(W_ 2=18\).
scientific article; zbMATH DE number 3895843