A locally flat embedding of a 
-dimensional manifold in an 
-dimensional manifold
is an embedding that, near each interior point
of its image, is homeomorphic to the standard inclusion 
. For a disk in a manifold
with boundary, the relative boundary condition means that its boundary
lies in the ambient boundary and its interior
lies in the ambient interior. At a boundary
point, local flatness requires the pair to have the half-space model 
, where 
and the inclusion pads with zeros (Friedl
et al. 2025, pp. 9-10). The relative boundary condition is additional
to being a proper map, which is automatic for a continuous embedding of
a compact set into a Hausdorff manifold. The condition concerns local topology, rather
than differentiability.
A smooth embedded disk 
is locally flat at its interior points
that lie in the ambient interior. If it satisfies the
relative boundary condition and meets the ambient boundary
transversely, it also has the half-space model there. Here transversality means that
the tangent spaces satisfy 
for every 
. A topological embedded disk can have wild points and fail to be locally flat. The disk
embedding conjecture asks for disjoint locally flat disks
from specified immersed disks and algebraic dual spheres
in four-dimensional manifolds.
