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:{{citation|title=Lectures on Dynamical Systems, Structural Stability, and Their Applications|first=Kotik K.|last=Lee|publisher=World Scientific|year=1992|isbn=9789971509651|page=109|url=https://books.google.com/books?id=saQC-ZqEYj8C&pg=PA109}}.

• The boundary of $A$ is a subset of the boundary of $M$. That is, $\backslash partial\; A\; \backslash subset\; \backslash partial\; M$.{{dubious|date=September 2021}}
• 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 $(U,\; \backslash phi)$ of $M$ such that $A\; \backslash cap\; U\; =\; \backslash phi^\{-1\}(\backslash mathbb\{R\}^m)$ where $m$ is the dimension {{nowrap|of $A$.}} For instance, in the category of smooth manifolds, this means that the embedding of $A$ must also be smooth.