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 $f:M^m\to N^n$ of manifolds is called an “immersion” if the derivative $f_{*p}:\mathcal{T}_pM\to\mathcal{T}_{f(p)}N$ is injective at every point $p\in M$ . Immediately we can tell that this can only happen if $m\leq n$ .

Notice now that this does not guarantee that $f$ itself is injective. For instance, if $M=\mathbb{R}^1$ and $N=\mathbb{R}^2$ , then we can form the mapping $f(t)=(t-t^3,1-t^2)$ . Using the coordinates $t$ on $M$ and ${x,y}$ on $N$ , we can calculate the derivative in coordinates:

$\displaystyle f_{*t}:\frac{\partial}{\partial t}(t)\mapsto(1-3t^2)\frac{\partial}{\partial x}(f(t))-2t\frac{\partial}{\partial y}(f(t))$

The second component of this vector is only zero if $t$ itself is, but in this case the first component is $1$ , thus $f_{*t}$ is never the zero map between the tangent spaces. But $f(1)=f(-1)=(0,0)$ , so $f$ is not injective in terms of the underlying point sets of $M$ and $N$ .

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 $(-2,0)$ and $(0,2)$ , there is no self-intersection in each interval.

There is another, more subtle pathology to be careful about. Let $M$ be the open interval $(0,2\pi)$ , and left $f(t)=(\sin(t),\sin(2t))$ . 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 $f$ 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 $f(M)\subseteq N$ — endowed with the subspace topology — is homeomorphic to $M$ itself by $f$ . 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.

April 18, 2011 - Posted by John Armstrong | Differential Topology, Topology

6 Comments »

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

Pingback by Fourth Linkfest | April 23, 2011 | Reply
[…] 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 | Reply
[…] 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 | Reply
[…] 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 | Reply

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

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