On p. 65 of John M. Lee's book Introduction to Smooth Manifolds, we find the following as part of Exercise 3.17:
Verify that $(\tilde{x}, \tilde{y})$ are smooth coordinates on $\mathbb{R}^2$, where
$$\tilde{x}=x; \;\;\tilde{y} = x^3+y.$$
What is meant by showing that these are global smooth coordinates? Are we just showing that $(x,y) \mapsto (x,x^3+y):\mathbb{R}^2\to \mathbb{R}^2$ is a diffeomorphism?
Thanks folks
Best Answer
Yes. Global coordinates on a manifold $M$ are coordinates that are globally defined, and yield a homeomorphism (diffeomorphism in the case of differentiable manifolds) between $M$ and an open subset of $\mathbb{R}^d$ (where $d$ is the topological dimension of the manifold). In other words, local coordinates whose coordinate patch is the entire manifold.
Not to be confused with globally defined local coordinates, which are maps $\varphi\colon M \to \mathbb{R}^d$ such that every $m\in M$ has a neighbourhood $U$ such that the restriction $\varphi\lvert_U$ is a coordinate chart, like $z \mapsto e^z$ for $\mathbb{C}$ [of course, global coordinates are also globally defined local coordinates, but in general not vice versa].
So you need to check that $(x,y) \mapsto (\tilde{x},\tilde{y})$ is indeed a diffeomorphism (easy), and that the differential operators $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial\tilde{x}}$ are different (not quite as easy, but easy enough still).