Extending Whitney's extension theorem: nonlinear function spaces

TL;DR

This paper extends Whitney's extension theorem to nonlinear, manifold-valued smooth functions on complex domains, establishing conditions for the restriction map to be a submersion and proving the existence of extension operators.

Contribution

It generalizes Whitney's extension theorem to nonlinear, manifold-valued functions on domains with corners or rough boundaries, including the construction of extension operators.

Findings

Restriction map is a submersion of locally convex manifolds.

Existence of extension operators for complex domains.

Applicable to domains with mild boundary irregularities.

Abstract

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where $C$ only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.