Precisely, what is the polar coordinate system?

Question

We all have seen the polar coordinate system back in high school. Recall that $S^2$ may be parametrized by $(\theta,\phi)$ with the correspondence $$\begin{align} x(\theta,\phi) &= \sin(\theta)\cos(\phi) \\ y(\theta,\phi) &= \sin(\theta)\sin(\phi) \\ z(\theta,\phi) &= \cos(\theta) \end{align}$$

where $\theta\in [0,\pi]$ and $\phi\in [0,2\pi)$. There's nothing complicated about this. However, as I began to study some differential geometry and manifold, I find this to be confusing.

Viewing $S^2$ as a $2$-dimensional manifold, we want to think of this parametrization as a coordinate chart $(U,\psi)$, where $\psi$ maps $(x,y,z)$ to the corresponding $(\theta,\phi)$ in $\Bbb R^2$. Then $\psi^{-1}$ would be our parametrization of the sphere.

However, the definition requires that $\psi$ is a diffeomorphism from the open set $U$ onto its image. Clearly $[0,\pi]\times [0,2\pi)$ is not open (this is related to the fact that there's no global chart for $S^2$). Yes, we may restrict to $U\subsetneq S^2$ but that is not my question.

In the language of manifold theory, what does it mean to talk about the global chart or the global parametrization of a sphere?

Even though I know that it's an abuse of languge, strictly speaking, but it appears a lot in the literature. For example, I am reading a book on Riemannian manifold and there's a part that says

We therefore introduce on $\Bbb R^d$ the standard polar coordinates $$ (r,\varphi^1,\dots,\varphi^{d-1}) $$ where $\varphi=(\varphi^1,\dots,\varphi^{d-1})$ parametrizes the unit sphere $S^{d-1}$. ... Express the (Riemannian) metric in polar coordinates ... in these coordinates at $0\in T_pM$, $0$ corresponds to $p\in M$, $$ g_{rr}(0)=1,\ g_{r\varphi}(0)=0. $$

Why are we allowed to use regular rules of taking derivative and such in this "global coordinate"? How do we rigorously justify the use of this global coordinate system?

Answer 1

You should think of this parametrization as a ramified cover of the $2$-sphere $\mathbf{S}^2$ by $\mathbf{R}^2$. Namely, instead of restricting it to get a bijection from $[0,\pi] \times [0,2\pi)$ onto $\mathbf{S}^2$ (which requires a non-canonical choice of endpoints), consider it as a quotient map $$\mathbf{R}^2 \rightarrow \mathbf{S}^2$$ which a calculation of the Jacobian matrix shows is ramified precisely over the points $(0,0,\pm 1) \in \mathbf{S}^2$, when $\theta$ is an integer multiple of $\pi$. Away from these points, the map is a covering projection, and so you obtain charts by taking local sections. You can think of polar coordinates on $\mathbf{R}^n$ in the same way.

Given a metric $g$ on the codomain of a differentiable map between manifolds, you can pull it back to the domain to obtain a metric except on the fibres over the ramified points; this is the formalization of expressing $g$ in polar coordinates. Without knowing precisely what conclusions you wish to draw about $g$ itself it is difficult to say more about how to use this formalism.