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, $\partial A=A\cap \partial M$ .
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 $(U,\phi )$ 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.

References

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

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Neat submanifold

See also

References