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


 * $$M$$ be a manifold with boundary, and
 * $$A$$ be a submanifold of $$M$$.

Then $$A$$ is said to be a neat submanifold of $$M$$ if it meets the following two conditions:
 * The boundary of $$A$$ is a subset of the boundary of $$M$$. That is, $$\partial A \subset \partial M$$.
 * Each point of $$A$$ has a neighborhood within which $$A$$'s embedding in $$M$$ is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space.

More formally, $$A$$ must be covered by charts $$(U, \phi)$$ of $$M$$ such that $$A \cap U = \phi^{-1}(\mathbb{R}^m)$$ 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.