Whitney Embedding Theorem

Whitney Embedding Theorem

In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney:

• The strong Whitney embedding theorem states that any smooth m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in Euclidean 2m-space, if m>0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of dimension m cannot be embedded into Euclidean (2m−1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).

• The weak Whitney embedding theorem states that any continuous function from an n-dimensional manifold to an m-dimensional manifold may be approximated by a smooth embedding provided m>2n. Whitney similarly proved that such a map could be approximated by an immersion provided m>2n−1. This last result is sometimes called the weak Whitney immersion theorem.