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, .[dubious –
discuss]
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.