Differential geometry: what are coordinates, and what is their relation to a basis set?

Question

I'm self-studying differential geometry (from several different places) and have run into what I think is a high-level conceptual confusion. As with most conceptual confusions, I am not exactly certain how to express it even, but I think it comes down to: what are coordinates, and what is their relation to a basis set?

As an engineer, likely my notation is terrible - I apologize and welcome any corrections/guidance on it, I'm trying to learn.

I've so far been thinking/reasoning from a vector space point of view. If we have a basis set $e=\{e_1, e_2...e_n\}$ for a vector space $V$, then we can write $v\in V$ as $v=v_1e_1+v_2e_2+...v_ne_n$ (ignoring covariant/contravariant components distinction at the moment). From what I understand $[v]_e=(v_1, v_2...v_n)$ are the coordinates of $v$ in the basis set $e$.

In DG, we want to see what happens when we "change the coordinate system" (haven't managed to find a formal definition of this anywhere yet, but of course I've done this in calculus before). I understand this to imply that we are actually changing basis. But in many places (e.g. page 8 in https://www.astro.gla.ac.uk/users/norman/lectures/GR/part3-screen.pdf) the basis vectors are defined in terms of coordinate curves, e.g.: $e_i=\frac{\partial}{\partial x_i}$ (where $x_i$ is the coordinate). However, based on my previously expressed reasoning, I would have thought that the coordinates are simply the coefficients needed for expressing a vector in the basis vector set (after the basis set has been selected). I know that we can probably go the other direction in the equation (could we?) and get the coordinate curves from a selected basis set, but from linear algebra, there are requirements of linear independence and span for the basis set. I haven't seen any textbook that explains how these requirements would translate to coordinate curves. So the basis set seems "more fundamental" than the coordinate curves to me (or at least have requirements for them).

Am I thinking about this in the wrong way for DG? Are the coordinate curves on the manifold the more important object from which we should define the basis?

Answer 1

In a nutshell, yes, you might be thinking it just a tad wrong.

The thing is, when you work in terms of linear algebra, you basically work within fixed vector spaces. That is, whenever you start a new chapter, that'll most likely include: "consider a vector space $V$, with the dimensionality of $n$ and a scalar product given by the rule $\varphi: V\times V \rightarrow \mathbb{R}$". And here, explicitly, you know that you have got a vector structure, hence you've got a basis, hence you can expand any vector within this vector space as a linear combination of basis vectors.

On the other hand, when you deal with differential geometry, especially with manifolds, you never get a vector structure in the first place and that's why manifolds matter. The sphere $\mathbb{S}^2$, as a manifold, exists regardless of the Euclidean space it can be embedded into.

Imagine this: an eternal creature creates a sphere and you on that sphere. Apart from anything, you don't know what it is to have a vector on a manifold, that is just outside of the scope of your comprehension. Now the first thing you do, is define the coordinates, i. e., you state the rules of assigning each point $P \in \mathbb{S}^2$ a specific set of numbers $(u_1^P, u_2^P, \dots, u_n^P)$. Once you have defined such numbers $\forall P$ in some neighbourhood, and those numbers change continuously from a point to a point, then you have defined the local coordinates.

As it happens on manifolds, prior to introducing the notion of a vector, you have to introduce the notion of coordinates. Once you have the latter, you unite the sequence of points in a path and get a curve on the manifold. That curve $\gamma(t): [a, b]\rightarrow M$ is a function of coordinates: $$\gamma(t) = [u_1(t), u_2(t), \dots, u_n(t)].$$

The rest is simple: you define a tangent vector $\xi$ at a point $P=\gamma(t_0)$ as a velocity vector of the curve: $\xi = \frac{d}{dt}\gamma(t)\big|_{t_0}$. It is only now that you have created a connection between a manifold and some sort of a vector structure.

After that you prove that $\partial_{u_1}, \partial_{u_2}, \dots, \partial_{u_n}$ form a basis in $T_{P}M$ and conduct all the analysis with tangent spaces / bundles and such.