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Given a metric of the form $$ds^2=dr^2+a^2\tanh^2(r/b)d\theta^2$$ why does it follow that $a=b$? I can't quite spot a constraint condition...

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Essentially a duplicate of physics.stackexchange.com/q/54627/2451 – Qmechanic♦ Feb 22 at 10:49
There are no constraints on a metric apart from ones your particular problem would impose on it. So we need more context. – Michael Brown Feb 22 at 10:53
@Qmechanic: Ahhh! Thanks! – Bert Feb 22 at 10:57

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Maybe so you'd recover $$ds^2=dr^2+r^2d\theta^2$$ at small $r$. I think that we need more details about this metric (what are its motivations/applications)

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That's right, the condition $a=b$ follows from the smoothness at $r\to 0$. For $a\neq b$, there would be a deficit or excess angle - the geometry would be a cone with a sharp singular tip at $r=0$. – Luboš Motl Feb 22 at 11:09
Thank you, Learning is a mess and @LubošMotl. – Bert Feb 22 at 14:00

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