Now, this is kind of embarrassing, but I've been trying to do this for too long now and failed: I want to construct an atlas for ##S^2##, but I want to use spherical coordinates rather than stereographic projection.(adsbygoogle = window.adsbygoogle || []).push({});

Of course the first chart image is simply ##\theta \in (0, \pi), \varphi \in (0,2 \pi)##, which is in a sense the sphere without a line along ##\varphi = 0## going from ##\theta =0## to ##\theta = \pi##.

All I want now is a second chart that is in a sense "complementary" to the first one, that is I want to cut the sphere along the ##\theta= \pi /2##, ##\varphi \in ( \pi/2 , 3 \pi /2)## line and use a spherical coordinate system for the rest with a simple transition function that just shifts the angles. Is this even possible? If so, how?

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# Atlas on S^2

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