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Neat submanifold

From Wikipedia, the free encyclopedia

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

be a manifold with boundary, and
be a submanifold of .

Then is said to be a neat submanifold of if it meets the following two conditions:[1]

  • The boundary of is a subset of the boundary of . That is, .[dubiousdiscuss]
  • Each point of has a neighborhood within which 's embedding in is equivalent to the embedding of a hyperplane in a higher-dimensional Euclidean space.

More formally, must be covered by charts of such that where is the dimension of . For instance, in the category of smooth manifolds, this means that the embedding of must also be smooth.

See also

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References

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