Whitney embedding theorem

Any smooth real m-dimensional manifold can be smoothly embedded in real 2m-space

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 real m-dimensional manifold can be smoothly embedded in the real 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 real (2m − 1)-space if m is a power of two. 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 Whitney immersion theorem.

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