On submanifolds of two manifolds (Q1323453)

Suppose \(Q_ c^ N\) denotes the \(N\)-dimensional, complete and simply connected Riemannian manifold of constant sectional curvature \(c\). Consider an \(n\)-dimensional Riemannian manifold which admits two isometric immersions \(f: M\to Q^ N_ c\) and \(g: M\to Q_{\tilde c}^{\tilde N}\) with \(c\neq\tilde c\). Under assumptions on codimensions and on the second fundamental forms of \(f\) and \(g\), the authors describe the relation holding between \(f\) and \(g\) on an open dense subset on \(M\). This result is used to obtain a rigidity theorem for minimal immersions and also to generalize a previous theorem of the authors [J. Differ. Geom. 36, No. 1, 1-18 (1992; Zbl 0768.53027)].