Relations between a manifold and its focal set

Summary Throughout this paper, smooth meansC . All manifolds and embeddings will be smooth. By aclosed m-manifold we mean a compact connected manifold of dimensionm, without boundary.LetM be a closedm-manifold (m>0), andf: ME ⁿ an embedding in Euclideann-space. The focal points off are the centres of principal curvature (with respect to some normal direction) of the embedded manifoldf(M). These points form thefocal set C(f) off.The starting point for our investigation is the following problem. Is there any relation between the topological structure ofM and the relative positions ofC(f) andf(M) inE ⁿ ? In particular, canf be so chosen thatC(f) andf(M) are disjoint? We say that such an embedding isnonfocal.We find that there are manifolds for which no such embedding exists.