If you are given a metric g(adsbygoogle = window.adsbygoogle || []).push({}); _{αβ}and you are asked to find if it describes a flat space, is there any way to answer it without calculating the Riemman Tensor R^{λ}_{μνσ}?

and how can I find for that given metric the coordinate transformation which brings it in conformal form?

For example I'll give you the way I'm thinking on my problem.

The metric I'm given is in :

ds^{2}= exp(ax+by) (-dx^{2}+dy^{2})

If I want to show that the space is flat, I would compute the Riemman tensor and so it vanishes... (is there a faster way to do it?).

Then in order to find the transformation that would bring it in obvious conformal form, I would say that I want to write:

ds^{2}=exp(w(x',y')) (dx'^{2}+dy'^{2})

(translated to g_{αβ}=λ^{2}n_{αβ})

Is that a correct approach?

In the way that will it give me the x'(x,y) and y'(x,y) form?

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# Conformal form of metric

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