Can we provide the set $\{(x,y,z)\in\mathbb{R^3}|x^2+y^2=1\}$ with a 2-dimensional manifold structure involving only 1 chart? I can see it with 2 charts with cylindrical coordinates, but not with only one...
Looking for an atlas with 1 chart
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2iAccording to your definition, Is a chart a map from $M$ to any open subset of $\mathbb{R}^2$? Or is is a map from $M$ to any open *ball* of $\mathbb{R}^2$? – 2012-11-23
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0@Jason: I'll go for any open subset...what kind of difference does it make in the global theory? – 2012-11-23
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1Well, Hans's answer now handles it. If you restrict to open balls, then you need at least 2 charts. If you restrict to general open subsets of $\mathbb{R}^2$ an annulus (or a unit disc with origin removed) works. – 2012-11-23