Neat submanifold

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:^[1]

• The boundary of the submanifold coincides with the parts of the submanifold that are part of the boundary of the larger manifold. That is, ${\displaystyle \partial A=A\cap \partial M}$.
• 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 ${\displaystyle (U,\phi )}$ of $M$ such that ${\displaystyle A\cap U=\phi ^{-1}(\mathbb {R} ^{m})}$ 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.

See also

• Local flatness

References

1. Lee, Kotik K. (1992), Lectures on Dynamical Systems, Structural Stability, and Their Applications, World Scientific, p. 109, ISBN 9789971509651.