Some isometric minimal immersions of the three-dimensional sphere into spheres
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 \(I^ \# V(2, s)\).
scientific article; zbMATH DE number 737429