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A Whitney Extension Problem for Manifolds

Kevin O'Neill

Abstract

The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set $E\subset\mathbb{R}^n$, how can we tell if there exists a $d$-dimensional, $C^m$-smooth manifold $\mathcal{M}\supset E$? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when $d=1$.

A Whitney Extension Problem for Manifolds

Abstract

The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set , how can we tell if there exists a -dimensional, -smooth manifold ? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when .
Paper Structure (10 sections, 28 theorems, 194 equations)

This paper contains 10 sections, 28 theorems, 194 equations.

Table of Contents

  1. Introduction
  2. Background Material
  3. Basic Notation
  4. Manifolds and Transversality
  5. Solution to the Classical Whitney Extension Problem
  6. Glaeser Refinement for Manifolds
  7. Some Technical Lemmas
  8. Constructing Local Interpolants
  9. Pasting Local Interpolants Together
  10. On Continuous Selections

Key Result

Lemma 2.1

Let $d,n\in\mathbb{N}$. There exist constants $C=C(d,n)$ and $\epsilon_0=\epsilon_0(d,n)>0$ such that whenever $W,W'\in \mathop{\mathrm{Gr}}\nolimits(n,d)$ satisfy $\text{dist}(W,W')\le\epsilon\le \epsilon_0$, there exists $Q\in O(n)$ such that $QW'=W$ and $\|Q-I_n\|\le C\epsilon$.

Theorems & Definitions (50)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2: Termination Lemma
  • ...and 40 more