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I have some doubts about the definition of the chart of manifold applied to the sphere $ S^2 $ embedded in $ R^3 $. In every document I read, the example chart for the sphere is obtained using the stereography projection. No one takes as an example the inverse of the parametrization (for example the sphere of radius = 1)

$$\begin{eqnarray} x = \sin( \theta)\cos( \gamma )\\ y = \sin( \theta)\sin( \gamma )\\ z = \cos (\theta)\\ \end{eqnarray}$$

Why nobody cites this example? Isn't it a chart?

The only reason I may guess is the fact that in order to know the values of $\gamma$,$\theta$ corresponding to a point on a sphere the center of the sphere is needed and this is an element outside the surface of the sphere. Am I correct?

thank you

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In do Carmo's book, Differential Geometry of Curves and Surfaces, page 56, he uses this parametrization (inverse function of a chart).

In fact, this is a parametrization, and you can use this function on $\theta\in (0,2\pi)$ and $\gamma\in (0,\pi)$ in order to make this defined on an open set, a requirement parametrizations and charts.

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