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since my GR tutor can't help me with some issues arising I thought it is time to register here.

I am very confused about the phrase "coordinate independence". Especially regarding the Lie Derivative and the Commutator of two vector fields.

1)

The Lie Derivative is said to be coordinate independant, right?

Let's have a look at [tex]\mathcal{S}^2[/tex]. I know that I need at least 2 charts for it, but lets stick to usual polar coordinates

[tex]x = \cos \phi \sin \theta[/tex]

[tex]y = \sin \phi \sin \theta[/tex]

for identifying points on the sphere with points in [tex]\mathbb{R}^2 \supset [0,2 \pi]\times[0, \pi][/tex]

Then [tex]g_{ij} = diag(1, \sin^2 \theta)[/tex], taking the 1-coordiante to be [tex]\theta[/tex]

Now, we know that [tex]\mathscr{L}_{\partial_\phi} g = 0[/tex] but [tex]\mathscr{L}_{\partial_\theta} g \neq 0[/tex]

I am pretty sure (I didn't check it though) that if I was doing some coordinate change now, and I was changing the components of the metric tensor accordingly to [tex]g'_{ij}[/tex] and changing the vectors too, I would still get [tex]\mathscr{L}_{\parital'_\phi} g' = 0.[/tex]

This looks good then and one is tempted to say that we checked that the Lie derivative is independent of coordinates for an example.

BUT what confuses me now is that somehow the coordinates have been important here. Right from the beginning. We have chosen some polar axis to define our coordinate chart. We broke the symmetry by doing that, I guess, but I don't know what this actually means.

In my mind I imagine a sphere, perfectly round, no axis, no broken symmetry. Now I choose an axis. I now know that the corresponding "perpendicular" vectors [tex]\partial_\phi[/tex] are killing vectors.

Now I remember these killing vectors but begin with another axis. In these coordinates the vector looks different but the metric tensor does not. Hence the Lie derivative won't be still 0. It changed although we didn't change the manifold (the nice sphere in the head) and the vector (also fixed in the head). In this sense it is not coordinate independent.

What has happened here?

So I know that the coordinates we have chosen are "ill" because we just need one single map, but I can't imagine this alone gives rise to my confusion. For example one has to choose an axis for a stereographic projection as well.

Still, a bonus question would be, what problems _are_ arising due to this one-chart-are-no-atlas-problem.

2)

Let's say Lie Derivative and thus the commutator of 2 vector fields are coordinate independent.

I take an arbitrary point out of my manifold. I want to know the commutator of two given fields. I can choose any coordinates. I choose the coordinates in such a way that the 2 vectors of the 2 fields in this point of the manifold point in the same direction as the coordinate axes.

So the vectors are just partial derivatives. But this means the commutator vanishes. But this means every commutator vanishes everywhere if I do this for every point (assume the fields to be smooth).

Am I right, that the flaw here is, that the vector_fields_ are not just partial derivatives everywhere in the chart but just in one point. And I have to take the derivative first and then put in the point. So basically my argumentation would also mean that the ordinary derivative of every function from R to R is zero in every point because in a point the function is just a constant number.

The thing is, that the upper argumentation was used in some proof in the lecture to make a commutator vanish. But probably the vector fields have been special.

Is there a common case where one might use a similar argumentation? Or is there not, meaning the professor made a mistake?

Thank you very much for clearing up the confusions. Struggling with them for 2 weeks now :)