Existence of a coordinate system

Question

How can we formally show that a coordinate system $(x,y)$ exists or does not exist? For instance for some given coordinate system $(r,\phi,\theta)$ defined on the manifold $M =(1,\infty)\times\mathbb{S}^2$, does there exist a coordinate system $(s,\phi,\theta)$ for some tensor: $$h=ds⊗ds+g(s)^2(d\phi⊗d\phi+\cos^2\phi \space d\theta⊗d\theta)$$ where $g(s)$ is a positive smooth function.

NOTE: There is some $2$-tensor given on $M$ by: $$f=\frac{r}{r^3+r-2}dr \otimes dr +r^2d\phi \otimes d\phi + r^2\cos^2\phi \space d\theta\otimes d\theta$$

This is a problem I am stuck with for days already, just do not know the formal way of proving such coordinate systems exist, given some manifold $M.$ I would appreciate the help.

Your question is very unclear. Are you trying to find a change of coordinates $(r,\phi,\theta) \to (s(r,\phi,\theta),\phi,\theta)$ and a function $g$ so that $h=f$? — Anthony Carapetis, Apr 16 '18 at 02:00
Personally its just a quote direct from the question in my notes, and I think what you are saying makes more sense. — Aurora Borealis, Apr 16 '18 at 02:06
Essentially all we have to find is the correct $r$ in terms of $s$, such that it gives the tensor $h$ is it? — Aurora Borealis, Apr 16 '18 at 02:12
I don't know, you tell me! If I've correctly guessed the question, then the answer is to solve the differential equation $\frac{r}{r^3+r-2}{dr}^2 = {ds}^2$ for $s(r)$ and then set $g(s(r)) = r.$ — Anthony Carapetis, Apr 16 '18 at 02:22
$dr\otimes dr$ can be written as $dr^2?$ I can solve the d$r $ and $ds$ case, but how do you do it for $dr^2 $and $ds^2?$ — Aurora Borealis, Apr 16 '18 at 02:23

Anthony Carapetis · Accepted Answer · 2018-04-16T03:06:49.713

2

This answer assumes the interpretation of the question I proposed in the comments: given the tensor

$$h=\frac{r}{r^3+r-2}dr \otimes dr +r^2d\phi \otimes d\phi + r^2\cos^2\phi \space d\theta\otimes d\theta$$

defined on $M=(1,\infty)\times \mathbb S^2$ where $\theta,\phi$ are the standard spherical coordinates, find a change of coordinate $s=s(r)$ and a smooth function $g$ so that $$h=ds \otimes ds+g(s)^2(d\phi \otimes d\phi+\cos^2\phi \space d\theta\otimes d\theta).$$

We can immediately read off $r^2 = g(s)^2;$ so the hard part is finding the change of coordinate. Since $ds = \frac{ds}{dr}dr,$ equating the two expressions for $h$ given above yields $$\frac r{r^3 + r - 2}dr \otimes dr = \frac{ds}{dr}dr \otimes \frac{ds}{dr}dr$$ and thus $$\frac{ds}{dr} = \pm \sqrt{\frac{r}{r^3 + r - 2}}.$$ Since $r^3+r-2>0$ for $r>1,$ the RHS here is real with constant sign; so we can integrate it to find a general solution $s(r)$ which is injective and thus a valid change of coordinate:

$$s(r') = s(1) \pm \int_1^{r'}\sqrt{\frac{r}{r^3 + r - 2}}\,dr. $$ This integral has no expression in terms of elementary functions, so I would just leave it written like this; if you try a CAS you can get some horrible expression in terms of elliptic integrals.

edited Apr 16 '18 at 03:06

answered Apr 16 '18 at 02:53

Anthony Carapetis

35,361

O yes I was trying to solve this integral, but indeed it was hard. How can one show that $s(r)$ is injective? – Aurora Borealis Apr 16 '18 at 02:55
@AuroraBorealis: The differential equation tells you that $s'(r)$ is either strictly positive or strictly negative (depending on that $\pm$), and thus $s(r)$ is either strictly increasing or strictly decreasing. – Anthony Carapetis Apr 16 '18 at 02:56
Ok, and finally for the last line why is it $s(r') $ not $s(r)?$ – Aurora Borealis Apr 16 '18 at 02:56
To avoid overloading the variable $r$ by using it as both the variable of integration and one of the limits. – Anthony Carapetis Apr 16 '18 at 02:58
Ok but it does actually mean $s(r)$ is that correct? – Aurora Borealis Apr 16 '18 at 02:59
It's defining the function $s(r)$, yes. You can write it as $$s(r) = s(1) \pm \int_1^{r}\sqrt{\frac{x}{x^3 + x - 2}},dx$$ if you prefer. – Anthony Carapetis Apr 16 '18 at 03:00
Thank you this was very helpful. – Aurora Borealis Apr 16 '18 at 03:03
Your explanation was very helpful, could you maybe also take a look at my other problem, maybe tell me some directions which I have to take? https://math.stackexchange.com/questions/2736343/composition-of-maps, I would appreciate it. – Aurora Borealis Apr 16 '18 at 03:35

Existence of a coordinate system

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