Whitney embedding theorem

The Whitney embedding theorem is a fundamental theorem in differential geometry . It was proven in 1936 by the American mathematician Hassler Whitney . The theorem says that every -dimensional differentiable manifold has an embedding in . ${\displaystyle n}$ ${\displaystyle \mathbb {R} ^{2n}}$

Explanations

The key message of this theorem is that there are actually only manifolds in Euclidean space .

Note that the theorem is only valid if one follows the (very common) definition that a manifold is always two-countable . If this is not required, there are smooth manifolds that cannot be embedded in Euclidean space, such as B. the long straight line or an uncountable discrete space .

Embedding one manifold into another is an injective mapping , so that a submanifold is of and the mapping is a diffeomorphism. Put clearly, an embedding in Euclidean space results in a surface that nowhere penetrates or touches one another. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle f\colon M\to N}$ ${\displaystyle f(M)}$ ${\displaystyle N}$ ${\displaystyle f\colon M\to f(M)}$ ${\displaystyle \mathbb {R} ^{n}}$

example

One example is the Klein bottle , a two-dimensional manifold that cannot be embedded (but immersed ) in three-dimensional space , but it can in four- dimensional space . ${\displaystyle \mathbb {R} ^{4}}$

The example of the embedding of the torus in three-dimensional space shows that the dimension is not always the smallest dimension for which an embedding exists; sometimes a lower dimension is sufficient. But Whitney's result is sharp in the sense that for each there is a -dimensional manifold that can be embedded in -dimensional space but not in -dimensional space. ${\displaystyle 2n}$ ${\displaystyle n=2^{k}}$ ${\displaystyle n}$ ${\displaystyle 2n}$ ${\displaystyle (2n-1)}$

literature

• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2002, ISBN 0-387-95448-1 .

See also

• Nash embedding theorem
• Whitney's Immersion Theorem

Web links

• Classification of embeddings