On nonpositively curved Euclidean submanifolds: splitting results (Q1284458)

Let \(f\) be an isometric immersion of a Riemannian manifold \(M^{n}\) into Euclidean space \({\mathbb{R}}^{n+p}\). Denote by \(\nu(x)\) the dimension of the null space of the second fundamental form at \(x\). \(\nu\) is called the index of relative nullity of \(f\). It was shown by the first author of the paper [Math. Ann. 298, 187-192 (1994; Zbl 0810.53011)] that \(\nu \geq n-2p\) if \(M^{n}\) has nonpositive sectional curvature. In the present paper, the authors study immersions of minimal index. Namely, they prove that if \(M^{n}\) has nonpositive curvature and \(\nu=n-2p\), then, on an open dense subset of \(M^{n}\), \(f\) splits locally as a product of \(p\) nowhere flat Euclidean hypersurfaces of nonpositive sectional curvature. An analogous result for immersions into a (holomorphically) flat complex space form is also proved in the paper.