Warped product structure of submanifolds with nonpositive extrinsic curvature in space forms (Q884009)

Let \(f: M^n \mapsto Q_c^{n+p}\) be an \(n\)-dimensional isometrically immersed submanifold of codimension \(p\) in a space form \(Q_c^{n+p}\) with nonpositive extrinsic sectional curvature, that is, the sectional curvature \(K_M\) of \(M^n\) verifies \(K_M\leq c.\) The index of relative nullity \(\nu =\nu ^f\) of \(f\), i.e., the dimension of the nullity space \(\Delta \) of the second fundamental form \(\alpha \) of \(f,\) \[ \Delta =\text{Ker}\, \alpha =\{ X \in TM \mid \alpha (X, \cdot )=0\} \] satisfies \(\nu \geq n-2p\) everywhere. If \(\nu \) is bigger, then one gets more geometrical restrictions. In the present paper the author constructs several examples of submanifolds with nonpositive extrinsic curvature and minimal index of relative nullity in any space by means of a simple warped product construction. Then, this is used to extend splitting results known for Euclidean submanifolds with nonpositive sectional curvature to arbitrary space forms.