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 M be a manifold with boundary, and A a submanifold of M . A is said to be a neat submanifold of M if it meets the following two conditions:
The boundary of the submanifold coincides with the parts of the submanifold that are part of the boundary of the larger manifold. That is, ∂ A = A ∩ ∂ 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, A must be covered by charts ( U , ϕ ) of M such that A ∩ U = ϕ − 1 ( 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.