Neat submanifold

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 ${\displaystyle M}$ be a manifold with boundary, and ${\displaystyle A}$ a submanifold of ${\displaystyle M}$. A is said to be a neat submanifold of ${\displaystyle M}$ if it meets the following two conditions:[1]

• The boundary of the submanifold is a subset of the boundary of the larger manifold. That is, ${\displaystyle \partial A\subset \partial M}$.
• 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, ${\displaystyle A}$ 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.