Whitney immersion theorem
On immersions of smooth m-dimensional manifolds in 2m-space and (2m-1) space / From Wikipedia, the free encyclopedia
In differential topology, the Whitney immersion theorem (named after Hassler Whitney) states that for , any smooth
-dimensional manifold (required also to be Hausdorff and second-countable) has a one-to-one immersion in Euclidean
-space, and a (not necessarily one-to-one) immersion in
-space. Similarly, every smooth
-dimensional manifold can be immersed in the
-dimensional sphere (this removes the
constraint).
The weak version, for , is due to transversality (general position, dimension counting): two m-dimensional manifolds in
intersect generically in a 0-dimensional space.