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The book on general relativity I'm reading states that the motivation behind the concept of manifold is to extend the theory of analysis in $ℝ^n$ to curved spaces. As an example of manifold, it gives the 2-sphere $S^2$ \begin{equation} S^2 = \{x,y,z \in ℝ^3 | x^2 + y^2 + z^3 =1 \} \end{equation} and how it is possible to map this set with charts to $ℝ^2$

I don't understand why it is not enough to consider the parametrization of the sphere as:

\begin{equation} \begin{cases} x= R \sin \phi \cos \theta\\ y= R \sin \phi \sin \theta\\ z=R \cos \phi\\ \end{cases} \end{equation}

and use vectorial analysis instead of resorting to manifolds.

If it is a matter of generalization to all possible curved spaces then if it is true that a manifold is a continuous space which "looks" locally like Euclidean open subsets of $ℝ^n$, In the case of curved surfaces shouldn't be possible to approximate locally the surface by that of a spherical cap?

Michaelangelo Meucci
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  • Please, re-read your question: Is this really the equation of a sphere? What do you mean by "why it is not enough"? Enough for what? They are trying to explain what an abstract manifold is and the language of charts is superior to the one of equations and "parameterizations". What do you mean by a "continuous space"? (Topological space?) Etc. – Moishe Kohan May 11 '19 at 12:27

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No, it isn't possible to approximate locally a smooth surface with a spherical cap. On a spherical cap, the curves which are the intersections with normal planes at some point have all the same curvature. This is not the case for a generic surface.

Intelligenti pauca
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