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A Local Condition for Totally Skew Embeddings

Zachary Norfolk

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.

A Local Condition for Totally Skew Embeddings

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 , and to solve the totally skew embedding problem for with a power of 2. Some algebraic and geometric properties of this condition are also discussed.

Paper Structure

This paper contains 13 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Summary of Results
  3. Acknowledgements
  4. The Local Condition
  5. Proof of the Local Condition
  6. Spherical Blow-Up
  7. Extension of $\widetilde{F}$
  8. Proof of Theorem \ref{['lc']}
  9. Construction Using the Local Condition
  10. Properties of the Local Condition
  11. Geometric Interpretation
  12. Stratification
  13. Appendix

Key Result

Theorem 2.1

Suppose $Df_a$, $D^2f_a$, and $D^3f_a$ have the property that solutions $(v_1, v_2, v_3, \lambda) \in \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R}$ of the equation with $v_3 \neq 0$ must have $v_1 = 0, v_2 = 0$ and $\lambda = 0$. Then $f$ is a totally skew embedding when restricted to some open neighborhood of $a$.

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4: Taylor's Theorem
  • Lemma 2.5
  • proof
  • proof
  • ...and 10 more