title: Neat submanifold
text: # 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.
  
  