  We provide a natural generalization to submanifolds of the holographic method
used to extract higher-order local invariants of both Riemannian and conformal
embeddings, some of which depend on a choice of parallelization of the normal
bundle. Qualitatively new behavior is observed in the higher-codimension case,
giving rise to new invariants that obstruct the order-by-order construction of
unit defining maps. In the conformal setting, a novel invariant (that vanishes
in codimension 1) is realized as the leading transverse-order term appearing in
a holographically-constructed Willmore invariant. Using these same tools, we
also investigate the formal solutions to extension problems off of an embedded
submanifold.
