Topology/Geometry

General topology

In general topology: an embedding is a homeomorphism onto its image.

Differential geometry

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.

Riemannian geometry

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).

1. For all x₁, x₂ in X, x₁ ≤ x₂ if and only if F (x₁) ≤ F(x₂) and
2. For all y in Y, {x : F (x) ≤ y } is directed.