Neat submanifold

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

${\displaystyle M}$ be a manifold with boundary, and

${\displaystyle A}$ be a submanifold of ${\displaystyle M}$.

Then ${\displaystyle A}$ is said to be a neat submanifold of ${\displaystyle M}$ if it meets the following two conditions:^[1]

• The boundary of ${\displaystyle A}$ is a subset of the boundary of ${\displaystyle M}$. That is, ${\displaystyle \partial A\subset \partial M}$.^{[dubious – discuss]}
• Each point of ${\displaystyle A}$ has a neighborhood within which ${\displaystyle A}$'s embedding in ${\displaystyle M}$ is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space.

More formally, ${\displaystyle A}$ must be covered by charts ${\displaystyle (U,\phi )}$ of ${\displaystyle M}$ such that ${\displaystyle A\cap U=\phi ^{-1}(\mathbb {R} ^{m})}$ where ${\displaystyle m}$ is the dimension of ${\displaystyle A}$. For instance, in the category of smooth manifolds, this means that the embedding of ${\displaystyle A}$ must also be smooth.