  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 $d=1$.

Functional Analysis