Minimal immersions into space forms with two principal curvatures
Let \(Q_ c^{n+p}\) be an \((n+p)\)-dimensional space of constant curvature c and \(M^ n\) a Riemannian manifold of dimension \(n\geq 3\). The aim of this paper is to classify minimal immersions of M into \(Q_ c^{n+p}\) such that, for any normal direction \(\xi\), the shape operator \(A_{\xi}:\) T\(M\to TM\) has at most two distinct eigenvalues. For \(p=1\) \textit{T. Otsuki} proved that if A has no simple eigenvalues, then \(c>0\) and M is locally the product of two spheres [Am. J. Math. 92, 145-173 (1970; Zbl 0196.251)]. \textit{M. do Carmo} and \textit{M. Dajczer} proved that if one eigenvalue is simple, then M is a rotational hypersurface [Trans. Am. Math. Soc. (to appear)]. See also the above mentioned paper of Otsuki.) For \(p>1\) some partial results were obtained by \textit{Y. Matsuyama} [Math. Z. 175, 275-282 (1980; Zbl 0437.53046)]. We give a complete classification of such immersions proving that if \(p>1\) and the immersion is substantial, then \(c>0\), M is an open part of a projective plane over the complex, quaternion or Cayley numbers and the immersion is the standard embedding of such spaces in the sense of \textit{D. Ferus} [Math. Ann. 247, 81-93 (1980; Zbl 0446.53041)]. Moreover the standard embeddings of the above mentioned spaces verify our conditions.
scientific article; zbMATH DE number 3855938