differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

, an area of mathematics, a neat submanifold of a

manifold with boundary In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...

is a kind of "well-behaved"

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...

. To define this more precisely, first let :

M

be a manifold with boundary, and :

A

be a submanifold of

M

. Then

A

is said to be a neat submanifold of

M

if it meets the following two conditions:. *The boundary of

A

is a subset of the boundary of

M

. That is,

\partial A \subset \partial M

. *Each point of

A

has a neighborhood within which

A

's embedding in

M

is equivalent to the embedding of a

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

in a higher-dimensional Euclidean space. More formally,

A

must be

covered Cover or covers may refer to: Packaging * Another name for a lid * Cover (philately), generic term for envelope or package * Album cover, the front of the packaging * Book cover or magazine cover ** Book design ** Back cover copy, part of co ...

charts A chart (sometimes known as a graph) is a graphical representation for data visualization, in which "the data is represented by symbols, such as bars in a bar chart, lines in a line chart, or slices in a pie chart". A chart can represent tab ...

(U, \phi)

M

such that

A \cap U = \phi^(\mathbb^m)

where

m

is the dimension For instance, in the category of

smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

s, this means that the embedding of

A

must also be smooth.

References

Differential topology {{topology-stub

See also

References