## Immersions and Embeddings

As we said before, the notion of a “submanifold” gets a little more complicated than a naïve, purely categorical approach might suggest. Instead, we work from the concepts of immersions and embeddings.

A map of manifolds is called an “immersion” if the derivative is injective at every point . Immediately we can tell that this can only happen if .

Notice now that this does not guarantee that itself is injective. For instance, if and , then we can form the mapping . Using the coordinates on and on , we can calculate the derivative in coordinates:

The second component of this vector is only zero if itself is, but in this case the first component is , thus is never the zero map between the tangent spaces. But , so is not injective in terms of the underlying point sets of and .

Courtesy of Wolfram Alpha, we can plot this map to see what’s going on:

The image of the curve crosses itself at the origin, but if we restrict ourselves to, say, the intervals and , there is no self-intersection in each interval.

There is another, more subtle pathology to be careful about. Let be the open interval , and left . We plot this curve, stopping just slightly shy of each endpoint:

We see that there’s never quite a self-intersection like before, but the ends of the curve come right up to almost touch the curve in the middle. Going all the way to the limit, the image of is a figure eight, which includes the crossing point in the middle and is thus not a manifold, even though the parameter space is.

To keep away from these pathologies, we define an “embedding” to be an immersion where the image — endowed with the subspace topology — is homeomorphic to itself by . This is closer to the geometrically intuitive notion of a submanifold, but we will still find the notion of an immersion to be useful.

As a particular example, notice (and check!) that the inclusion map of an open submanifold, as defined earlier, is an embedding.

[…] both of our pathological examples last time, the problems were very isolated. They depended on two separated parts of the domain manifold […]

Pingback by Immersions are Locally Embeddings « The Unapologetic Mathematician | April 19, 2011 |

[…] subspace topology from — then we say that is a submanifold of if the inclusion map is an embedding. If the inclusion is only an immersion, we say that is an “immersed submanifold” of […]

Pingback by Submanifolds « The Unapologetic Mathematician | April 20, 2011 |

[…] Armstrong: The Implicit Function Theorem, Immersions and Embeddings, Immersions are locally […]

Pingback by Fourth Linkfest | April 23, 2011 |

[…] map, and every point is a regular value. Its preimage is a submanifold diffeomorphic to . The embedding realizing this diffeomorphism is . The tangent space at a point on the submanifold is mapped by […]

Pingback by The Tangent Space of a Product « The Unapologetic Mathematician | April 27, 2011 |

[…] it is surjective, and if every point is a regular point of . Despite the similarity of the terms “immersion” and “submersion”, these are very different concepts, so be careful to keep them […]

Pingback by Submersions « The Unapologetic Mathematician | May 2, 2011 |

[…] But this time it’s not going to be quite so general; we will only extend our notion of an embedding, and particularly of an embedding in codimension […]

Pingback by Orientation-Preserving Mappings « The Unapologetic Mathematician | September 1, 2011 |