# Neat submanifold

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### Large Gauge Symmetry in Non-Abelian Gauge Theory (Lecture 6 of 10)

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 be a manifold with boundary, and a submanifold of . A is said to be a neat submanifold of if it meets the following two conditions:The boundary of the submanifold is a subset of the boundary of the larger manifold. That is, . 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, 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.
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