Embeddings With Multiple Regularity

We introduce $(k,l)$-regular maps, which generalize two previously studied classes of maps: affinely $k$-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a $(k,l)$-regular map. The problem can be regarded as an extension of embedding theory to embeddings with certain non-degeneracy conditions imposed, and is related to approximation theory.