The Unapologetic Mathematician

Mathematics for the interested outsider

Open Submanifolds

Eek! None of these drafts went up on time!

In principle, we know what a submanifold should be: a subobject in the category of smooth manifolds. That is, a submanifold S of a manifold M should be another manifold, along with an “inclusion” map which is smooth and left-cancellable.

On the underlying topological space, we understand subspaces; first and foremost, a submanifold needs to be a subspace. And one easy way to come up with a submanifold is just to take an open subspace. I say that any open subspace S\subseteq M is automatically a submanifold. Indeed, if (U,\phi_U) is a coordinate patch on M, then (U\cap S,\phi_U\vert_{U\cap S}) is a coordinate patch on S. The intersection U\cap S is an open subset, and the restriction of \phi_U to this intersection is still a local homeomorphism. Since the collection of all coordinate patches in our atlas cover all of M, they surely cover S as well.

As a quick example, an open interval in the real line is automatically an open manifold of \mathbb{R}, and so it’s a manifold. Any open set U in any n-dimensional real vector space is also automatically an n-manifold.

More generally, it turns out that what we want to consider as a “submanifold” is actually somewhat more complicated, and we will have to come back to this point later.

March 7, 2011 - Posted by | Differential Topology, Topology


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