# The Unapologetic Mathematician

## 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.