A Local Condition for Totally Skew Embeddings

TL;DR

This paper introduces a new third-order differential condition that ensures submanifolds are totally skew, leading to improved embeddings of Euclidean spaces and solving the embedding problem for spaces with dimensions that are powers of two.

Contribution

The paper presents a novel differential condition for totally skew embeddings and applies it to construct better embeddings and solve the embedding problem for certain Euclidean spaces.

Findings

Constructed improved totally skew embeddings of ^n

Solved the totally skew embedding problem for ^n with n a power of 2

Discussed algebraic and geometric properties of the new condition

Abstract

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.