embedding (original) (raw)

Let M and N be manifolds and f:M→N a smooth map. Then f is an embedding if

1. 1. f⁢(M) is a submanifold of N, and
2. 1. f:M→f⁢(M) is a diffeomorphism. (There’s an abuse of notation here. This should really be restated as the map g:M→f⁢(M) defined by g⁢(p)=f⁢(p) is a diffeomorphism.)

The above characterization can be equivalently stated:f:M→N is an embedding if

1. 1. f is an immersion, and
2. 1. by abuse of notation, f:M→f⁢(M) is a homeomorphism.

Remark. A celebrated theorem of Whitney states that every n dimensional manifold admits an embedding into ℝ2⁢n+1.