A Local Condition for Totally Skew Embeddings

We introduce a third-order differential condition, analogous to nonzero torsion of a curve, which guarantees a submanifold of Euclidean space is totally skew in a small neighborhood. This condition is used to construct improved totally skew embeddings of ${\mathbb{R}}^n$, and to solve the totally skew embedding problem for ${\mathbb{R}}^n$ with $n$ a power of 2. Some algebraic and geometric properties of this condition are also discussed.