nLab Whitney embedding theorem

Idea

The (strong) Whitney embedding theorem states that every smooth manifold (Hausdorff and sigma-compact) of dimension $n$ has an embedding of smooth manifolds in the Euclidean space of dimension $2n$.

Notice that it is easy to see that every smooth manifold embeds into the Euclidean space of some dimension (this prop.). The force of Whitney’s strong embedding theorem is to find the lowest dimension that still works in general.

Related concepts

• The analogous statement for Riemannian manifolds and isometric embeddings is the Nash embedding theorem.

References

Named after Hassler Whitney.

• Lev Pontrjagin, Section 2.2 of: Smooth manifolds and their applications in Homotopy theory, Trudy Mat. Inst. im Steklov, No 45, Izdat. Akad. Nauk. USSR, Moscow, 1955 (AMS Translation Series 2, Vol. 11, 1959) (doi:10.1142/9789812772107_0001)

See also

• Wikipedia, Whitney embedding theorem

• Paul Rapoport, Introduction to Immersion, Embeddingand the Whitney Embedding Theorem, 2015 (pdf)