I have recently had discussions on related questions about coordinate charts on here which has started to clear up some issues in my understanding of manifolds. Apologies in advance for the long-windedness of this post, but there are several points that I feel are crucial for me to understand to further my studies of differential geometry, and as they are fairly closely related I thought it better to ask/check them in one post.
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Several of the notes/textbooks that I've been reading on the subject have mentioned about the inverse map of a coordinate map as giving a local parametrization to a point in a given patch on a manifold. By this is it meant that, given an $n$-dimensional manifold $M$ and a homeomorphism $\phi:U\subset M\rightarrow V\subset\mathbb{R}^{n}$ from a patch on the manifold $U\subset M$, then we can parametrize a point $p\in U$ via the inverse map $\phi^{-1}:V\subset\mathbb{R}^{n}\rightarrow U\subset M$?
More explicitly, if $\phi (p)=(x^{1},\ldots,x^{n})$ are the coordinates labelling $p$ in $\mathbb{R}^{n}$, then is it correct to say that one can describe a point in a given patch on the manifold directly via its parametrization with respect to its coordinates, i.e.
$$p=(\phi^{-1}\circ\phi)(p)=\phi^{-1}(\phi(p))=\phi^{-1}(x^{1},\ldots,x^{n})=(u^{1},\ldots,u^{n})$$
where $(u^{1},\ldots,u^{n})$ is the local parametrization of $p$ on $M$, and $u^{i}=u^{i}(x^{1},\ldots,x^{n})$ are functions whose domain is $\mathbb{R}^{n}$.If so, what really is the difference between parametrizations of points and their corresponding coordinates? (Is it simply that the coordinates are labels that allow one to distinguish individual points in a patch on the manifold, and then we can use these to parametrize the patch such that each value of the coordinates describes a given point on the manifold (in terms of the parametrization)).
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If I have understand this notion of parametrization correctly, then is the following discussion correct? If we take the example of a 2-sphere $S^{2}\subset\mathbb{R}^{3}$, then is the 2-tuple $(\theta,\phi)\in\mathbb{R}^{2}$ the coordinates of a point on the manifold (with the mapping defined by $p\mapsto (\theta,\phi)$) and its corresponding local parametrization on the manifold, $(\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta))\in S^{2}\subset\mathbb{R}^{3}$ (with the inverse mapping defined by $(\theta,\phi)\mapsto (\sin(\theta)\cos(\phi),\sin(\theta)\sin(\phi),\cos(\theta))$) ?
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From reading John Lee's books on smooth manifolds and Riemannian geometry (and from a previous discussion on here), I think it is correct to say that (when a metric is defined on the manifold) one can only use local Cartesian coordinates (or local Euclidean coordinates) to label points in a patch on a manifold if the curvature of the manifold is zero (i.e. it is "locally flat") as then there will exist a local isometry between the manifold between the manifold and flat Euclidean space.
Mathematically, if $(M,g)$ is locally flat (i.e. has vanishing local curvature) then there will be an isometry $\psi$ to an open set in $(\mathbb{R}^{n},\bar{g})$(where $g$ is the metric defined on the $n$-dimensional manifold $M$, and $\bar{g}$ is the Euclidean metric defined on $\mathbb{R}^{n}$).
(Would it also be correct to say that the Cartesian coordinate system is the identity map $\text{id}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ defined by $(x^{1},\ldots,x^{n})\mapsto (x^{1},\ldots,x^{n})$, such that the parametrization of each point is simply given by $(x^{1},\ldots,x^{n})$?)
Apologies for the lack of rigour, I am coming from a Physics background and I'm self-teaching myself differential geometry in order to gain a better understanding of general relativity.
Best Answer
A manifold is an abstract axiomatic notion. Therefore, points in the manifold $M$ may be very weird objects. They may be matrices, other manifolds, whatever you want. As you say, the charts allow us to label points in the manifold by actual points in actual euclidean space, with the goal of doing some geometry and calculus (which we don't know how to do in $M$ a priori). In this sense, the coordinates of a point $p\in M$ are the corresponding coordinates of the point in $\mathbb{R}^{n}$, and the parametrization of the points in $M$ is this process of labelling the points in $M$ by points in space, so it is the inverse of the chart map.
You are right, if your chart assings to a point in an open subset of the sphere the two angles say with a meridian and the equator, then the coordinates of the point are those angles (a point in $\mathbb{R}^{2}$, as you said), and the parametrization is the inverse image of this map, which then assigns to points in the plane points in the sphere. Sorry for being circular, but indeed, it parametrizes points in the manifold by points in the plane (in the broad sense of the word parametrizing, e.g. points in projective space parametrize 1-dimensional linear subspaces of the underlying vector space).
I don't agree with your conclusion here. As I mentioned above, the goal of parametrising the points in your abstract manifold by points in euclidean space is to be able to do some geometry and calculus. Therefore, labelling points like that can always be done, at least in order to get a local parametrization of points in your manifold, which can already be very useful. Another example: you can do differential calculus on smooth manifolds thanks to these parametrizations, even if $M$ is not locally flat. If your point is something like "the manifold looks locally like euclidean space if it is locally flat", then I also disagree: this is to me a subjective statement, which depends on the properties that you are considering. For example, a locally flat manifold $M$ may very well not be a vector space. But euclidean space is. So the question is always relative to the properties you are looking for at that moment.