Reduction of Codimension

Abstract

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.