In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold. More precisely, let
M
be a manifold with boundary, and
A
a submanifold of
M
. A is said to be a neat submanifold of
M
if it meets the following two conditions:
The boundary of the submanifold coincides with the parts of the submanifold that are part of the boundary of the larger manifold. That is,
∂
A
=
A
∩
∂
M
.
Each point of the submanifold has a neighborhood within which the submanifold's embedding is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space. More formally,
A
must be covered by charts
(
U
,
ϕ
)
of
M
such that
A
∩
U
=
ϕ
−
1
(
R
m
)
where
m
is the dimension of
A
. For instance, in the category of smooth manifolds, this means that the embedding of
A
must also be smooth.
