Is it possible to avoid the spherical polar coordinate singularity on $S^2$ by taking the coordinates as they originally are on $T^2$, i.e. ranging from $0$ to $2\pi$ mod $2\pi$? How would one incorporate this in practice? The position vector to a point on a unit $S^2$ is ${\bf r} = {\bf e}_{\rho}$, where ${\bf e}_{\rho}$ is a unit vector in the direction of increasing radius. With the usual single cover, ${\bf e}_{\theta}$, where $\theta$ indicates the azimuth, becomes ambiguous at the North and South poles. How exactly is this avoided with the double cover?
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Johnver
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double cover ? you mean the coordinate chart we use in Riemannian manifolds ? – reuns Jun 09 '16 at 23:27
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2I don't see that it is. Remember that $S^2$ is simply connected and has no connected double-cover. – Ted Shifrin Jun 09 '16 at 23:38
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@TedShifrin : what does it mean ? – reuns Jun 09 '16 at 23:52
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Yes I am talking about coordinate charts here. – Johnver Jun 10 '16 at 00:04
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Use a second coordinate chart, namely spherical coordinates centered on the $x$-axis (instead of the $z$-axis). – Lee Mosher Jun 10 '16 at 00:39
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Related: Why spherical coordinates is not a covering?. – Andrew D. Hwang Jun 23 '16 at 11:49