Neat submanifold explained
In differential topology, an area of mathematics, a neat submanifold of a manifold with boundary is a kind of "well-behaved" submanifold.
To define this more precisely, first let
be a manifold with boundary, and
be a submanifold of
.
Then
is said to be a neat submanifold of
if it meets the following two conditions:
[1]
is a subset of the boundary of
. That is,
\partialA\subset\partialM
.
has a neighborhood within which
's embedding in
is equivalent to the embedding of a
hyperplane in a higher-dimensional Euclidean space.
More formally,
must be
covered by charts
of
such that
where
is the dimension For instance, in the category of smooth manifolds, this means that the embedding of
must also be smooth.
See also
Notes and References
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