# Tuesday, August 25

## Submanifolds

:::{.exercise}
Prove that charts on a manifold are smooth maps.

> Hint: use the identity smooth structure on $\RR^n$.

:::

:::{.exercise}
Show that open subsets of manifolds are again manifolds in a canonical way.
:::

:::{.exercise}
Show that $S^1$ is a manifold.
:::

:::{.example}
Prove that a submanifold is again a manifold.
:::