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What is the form of the space described by the metric $$ds^2=dr^2+k^2\,\,d\theta^2$$ where $k$ is a constant?

I believe this form of metric representation is in the so called geodesic polar coordinates.

I know that it cannot be Euclidean because that would require $r^2$ in place of $k^2$.

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If k is constant this is just the Euclidean metric with one of the directions rescaled. Set $ x=r$ and $y=k\theta$.

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Are you sure about this. The map $(r,e^{i\theta}) \mapsto (r,\theta)$ yields an isometry of $ds^2$ and the flat cylinder $]0,R[ \times S^1_{k}$, the latter being the circle of radius $k$. – wspin Feb 22 '13 at 12:46
As far as the metric goes, the two look the same. Local considerations don't tell you whether the $\theta$ direction is compact. – David Clarke Mar 1 '13 at 5:12
of course you are right, thanks – wspin Mar 1 '13 at 14:37

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