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.