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 {\displaystyle M} be a manifold with boundary, and
A {\displaystyle A} be a submanifold of M {\displaystyle M} . Then A {\displaystyle A} is said to be a neat submanifold of M {\displaystyle M} if it meets the following two conditions:
The boundary of A {\displaystyle A} is a subset of the boundary of M {\displaystyle M} . That is, ∂ A ⊂ ∂ M {\displaystyle \partial A\subset \partial M} . Each point of A {\displaystyle A} has a neighborhood within which A {\displaystyle A} 's embedding in M {\displaystyle M} is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space. More formally, A {\displaystyle A} must be covered by charts ( U , ϕ ) {\displaystyle (U,\phi )} of M {\displaystyle M} such that A ∩ U = ϕ − 1 ( R m ) {\displaystyle A\cap U=\phi ^{-1}(\mathbb {R} ^{m})} where m {\displaystyle m} is the dimension of A {\displaystyle A} . For instance, in the category of smooth manifolds, this means that the embedding of A {\displaystyle A} must also be smooth.
See also Local flatness
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