Whitney embedding theorem

The embedding theorem of Whitney is a fundamental theorem in differential geometry. It was proved in 1936 by the American mathematician Hassler Whitney. The theorem states that every -dimensional differentiable manifold possesses an injective immersion in. If the manifold is compact, it even has a closed embedding in.

Notes

The core message of this sentence, then, is that there are actually only manifolds in Euclidean space.

An embedding of a manifold to another is an injective mapping whose differential df is also injective. To put it clearly gives an embedding into the Euclidean space an area that pervades nowhere or touched.

Example

An example is the Klein's bottle, a two-dimensional manifold, which can not be embedded in three-dimensional space (but immersieren ), but in the four-dimensional.

The example of embedding of the torus in the three-dimensional space shows that the dimension is not always the smallest dimension, for which there is an embedding; sometimes a lower dimension is sufficient. But the result of Whitney is sharp in the sense that there is for each one -dimensional manifold that can be embedded in the -dimensional space, but not in the -dimensional space.