From Wikipedia, the free encyclopedia.
In mathematics, an embedding is one instance of some mathematical object contained within another instance, such as a group that is a subgroup.
In general topology: an embedding is a homeomorphism onto its image.
In differential geometry:
Let M and N be smooth manifolds and be a smooth map, it is called an
immersion if for any point the differential is injective (here denotes tangent space of at ).
An embedding, or smooth embedding, is an injective immersion.
In other words embedding is diffeomorphism to its image (in particular image of embedding is a submanifold). Immersion is a local embedding (i.e. for any point there is a naighborhood such that is an embedding.)
An important case is N=Rn. The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough. For example the real projective plane of dimension 2 requires n = 4 for an embedding. The less restrictive condition of immersion applies to the Boy's surface - which has self-intersections.
In Riemannian geometry:
Let (M,g) and (N,h) be Riemannian manifolds.
An isometric ebedding is a smooth embedding which preserves the Riemannian metric, i.e.
for any two tangent vectors we have . Equivalently, isometric embedding is a smooth embedding which preserve length of curves (cf. Nash embedding theorem).
Topology/Geometry
General topology
Differential geometry
Riemannian geometry
Domain theory
In domain theory, an embedding of partial orders is F in the function space [X → Y] such that
Based on an article from FOLDOC, used by permission.
