Given a smooth closed manifold \(M\), a smooth map \(f:M\to\mathbb R^p\) with \(n\geq p\geq 1\) is called a fold map it its only singular points are fold points. Fold singularities are the simplest type of stable singularities which these maps may have, and, therefore, fold maps can be seen as a generalization of Morse functions. For fold maps, the singular set is a regular closed submanifold of \(M\) of dimension \(p-1\) and \(f\) restricted to its singular set is an immersion.
In this paper the authors study round fold maps, which are fold maps where the map restricted to the singular set is an embedding onto a disjoint union of concentric spheres in \(\mathbb R^p\). These maps have nice properties such as being simple, i.e. each component of the preimage of a point in \(\mathbb R^p\) contains at most one singular point.
In particular, they succeed in characterizing which smooth closed \(n\)-manifolds admit round fold maps into \(\mathbb R^{n-1}\) for \(n\geq 4\). They also classify round fold maps up to \(\mathscr A\)-equivalence (smooth changes of coordinates in source and target).