Groups of motions and minimal immersions of spheres into spheres (Q1085821)

The author is investigating the space \(W_ 2\) defined by \textit{M. P. doCarmo} and \textit{N. R. Wallach} [Ann. Math., II. Ser. 93, 43-62 (1971; Zbl 0218.53069)]. Let f be an isometric minimal immersion of \(S^ m(1)\) into \(S^{n-1}(r)\). Let K be a skew \((m+1)\times (m+1)\) matrix and k the one-parameter subgroup of \(SO(m+1)\) generated by K. The element C of \(W_ 2\) associated with f is said to be K-invariant or k-invariant if \(f\cdot k\) is equivalent to f. The author extends this notion as follows. Let \(K_ 1,...,K_ p\) be skew matrices and k be the subgroup of \(SO(m+1)\) generated by those skew matrices. If C is invariant by \(K_ 1,...,K_ p\), C is said to be k- invariant. In the paper, the author especially studies invariant elements of \(W_ 2\) for \(m=3\) and \(s\geq 4\). This paper also contains some theorems on geodesics in minimal immersions. One of them is stated as follows. Let f be an isometric minimal immersion of \(S^ 3(1)\) into \(S^{24}(r)\), \(r^ 2=1/8\), such that the element C associated with f is g-invariant where g is an element of SO(4). Then, for any geodesic \(\gamma\) of \(S^ 3(1)\), the geodesic \(i\cdot f(\gamma)\) and \(i\cdot f(g\gamma)\) have the same set of curvatures \(k_ 1\), \(k_ 2\), \(k_ 3\), where i is the isometric embedding of \(S^{24}(r)\) into \(R^{25}\).