A Whitney Extension Problem for Manifolds

2023/10/19 by O'Neill, Kevin · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2310.13115

Abstract

The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set E⊂ℝⁿ, how can we tell if there exists a d-dimensional, C^m-smooth manifold M⊃ 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.

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• Computability of a Whitney Extension

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