Minimal isometric immersions of spherical space forms in spheres
A number of authors have studied minimal isometric immersions of Riemannian manifolds into round spheres, and in particular of round spheres into round spheres. As was observed by T. Takahashi, if \(\Phi: M \to S^ N(r) \subset \mathbb{R}^{N+1}\) is such a minimal immersion, then the components of \(\Phi\) must all be eigenfunctions of the Laplace operator on \(M\) corresponding to the same eigenvalue. The converse is correct. \textit{M. do Carmo} and \textit{N. Wallach} [Ann. Math., II. Ser. 93, 43-62 (1971; Zbl 0218.53062)] showed that in higher dimensions there are in general many minimal isometric immersions of \(S^ n(1)\) into \(S^ N(r)\), and they are parametrized by a compact convex body in a finite- dimensional vector space. In the present paper the following theorem is proved. Every homogeneous spherical space form admits a minimal isometric embedding into a standard sphere of appropriate radius. Half of the paper is devoted to the case of \(S^ 3\) separated from the rest because \(S^ 3\) is itself a group. The homogeneous three-dimensional space forms can be written as \(S^ 3/\Gamma\) where \(\Gamma\) is an arbitrary finite subgroup of \(S^ 3\). Thus the following groups are considered. The cyclic groups \(C_ d\). The binary dihedral groups \(D^*_ d\) and the groups \(T^*\), \(O^*\), \(I^*\). To summarize the results a table is presented. Some pages are devoted to higher-dimensional lens spaces and space forms of dimension \(4n-1\).
scientific article; zbMATH DE number 86166