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

[math]\displaystyle{ M }[/math] be a manifold with boundary, and

[math]\displaystyle{ A }[/math] be a submanifold of [math]\displaystyle{ M }[/math].

Then [math]\displaystyle{ A }[/math] is said to be a neat submanifold of [math]\displaystyle{ M }[/math] if it meets the following two conditions:

• The boundary of [math]\displaystyle{ A }[/math] is a subset of the boundary of [math]\displaystyle{ M }[/math]. That is, [math]\displaystyle{ \partial A \subset \partial M }[/math].
• Each point of [math]\displaystyle{ A }[/math] has a neighborhood within which [math]\displaystyle{ A }[/math]'s embedding in [math]\displaystyle{ M }[/math] is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space.

More formally, [math]\displaystyle{ A }[/math] must be covered by charts [math]\displaystyle{ (U, \phi) }[/math] of [math]\displaystyle{ M }[/math] such that [math]\displaystyle{ A \cap U = \phi^{-1}(\mathbb{R}^m) }[/math] where [math]\displaystyle{ m }[/math] is the dimension of [math]\displaystyle{ A }[/math]. For instance, in the category of smooth manifolds, this means that the embedding of [math]\displaystyle{ A }[/math] must also be smooth.

See also

• Local flatness