Embeddings with multiple regularity (Q884768)

The author introduces the definition of \((k,l)\)-regular maps. This notion is a generalization of totally skew embeddings and affinely \(k\)-regular maps. Remember that an immersion \(f:M^n\rightarrow\mathbb{R}^N\) is totally skew if for all points \(x\), \(y\) of \(M^n\) the tangent spaces \(df(T_xM)\) and \(df(T_yM)\) as affine subspaces of \(\mathbb{R}^N\) have an affine span of maximal possible dimension that is \((2n+1)\) and a continuous map \(f:X\rightarrow\mathbb{R}^N\) is called affinely \(k\)-regular if the images under \(f\) of \(k\) distinct points are affinely independent. \(K\)-regular maps are related to \(k\)-interpolating in approximation theory. In this article the author presents explicit examples of this type of maps and obtains bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a \((k,l)\)-regular map.