The study of isometric immersions becomes increasingly difficult for higher values of the codimension. Therefore, it is important to investigate whether the codimension of an isometric immersion into a space of constant sectional curvature can be reduced. That an isometric immersion \(f\colon M^n\to \mathbb {Q}_c^{n+p}\) admits a reduction of codimension to q < p means that there exists a totally geodesic submanifold \(\mathbb {Q}_c^{n+q}\) in \(\mathbb {Q}_c^{n+p}\) such that \(f(M)\subset \mathbb {Q}_c^{n+q}\). The possibility of reducing the codimension fits into the fundamental problem of determining the least possible codimension of an isometric immersion of a given Riemannian manifold into a space of constant sectional curvature.