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I just want to ask for a suggestion on the study of differential geometry. When I study it I understand the theorems, their proofs, I understand perfeclty the concepts and so forth, but I'm having trouble to find charts for manifolds. Can you give me a suggestion where to begin getting practice on it? And when I say begin, it's really begin, because even for trivial manifolds like $S^2$ I'm having a little trouble.

I know it's perhaps a silly question, however I think that other people may have the same problem when studying differential geometry and so I thought it was relevant to post it here. I'm really sorry if the question is too silly.

Thanks a lot in advance for your aid.

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    $\begingroup$ Take a look at John Lee's "Introduction to Smooth Manifolds." It's really a great book for beginners, and there are many examples of atlases provided there. $\endgroup$ Dec 1, 2012 at 7:57

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Well, it quite depends on how the manifold $M$ is constructed and presented to you.

A manifold of dimension $m$ should look locally like $\mathbb{R}^n$ and so you should be able to describe a neighborhood of $M$ by $m$ "parameters". If $M$ is given as a subset of $\mathbb{R}^n$ with the induced topology, you can put charts on $M$ by describing small neighborhoods of $p \in M$ as graphs of smooth functions. That is, if you suspect that $M$ is $m$ dimensional, you can provide for each $p \in M$ a neighborhood $U \subset \mathbb{R}^n$ such that $M \cap U$ is described by a permutation of something of the form $$ (x_1, ..., x_m, f_1(x_1, ..., x_m), ..., f_{n-m}(x_1, ..., x_m)). $$

This description above is valid if $U \cap M$ can be presented as a graph of a function over the variables $(x_1, ..., x_m)$ which you then use as the parameters. If it is a graph over some other set of variables $(x_{i_1}, ..., x_{i_m})$, you describe it accordingly with the other variables being dependent on $(x_{i_1}, ..., x_{i_m})$. This method for $S^2$ results in six charts that cover $S^2$ and is demonstrated in the cover of Do-Carmo's Riemannian Geometry:

enter image description here

It is of course not optimal with regard to the number of charts used. As $S^2 \setminus \{ \text{pt} \}$ is homeomorphic to $\mathbb{R}^2$, you expect to be able to describe it as a function of two parameters, which can be done using the stereographic projection. Then, you cover $S^2$ with two charts using the stereographic projections and check that the transition maps are smooth.

For another example, consider the infinite cylinder. If it is given to you as a product $M = S^1 \times \mathbb{R}$, you can simply use products of charts for $S^1$ and $\mathbb{R}$ to provide the charts. If, however, you are given the cylinder as a subset of $\mathbb{R}^3$ such as $$ M = \{ (\cos(\theta), h, \sin(\theta) \; | \; h, \theta \in \mathbb{R} \} $$ you can "identify" the product structure and use it to parametrize $M$.

If $M = M' / \sim$ is a quotient space of another manifold $M'$ (such as the case of $\mathbb{RP}^n$ or $\mathbb{CP}^n$), you can try to understand how the charts on $M'$ project to $M$. If $M$ is a linear Lie group, you can use sometimes some linear algebra decompositions and tricks to parametrize small neightborhoods of $M$, etc, etc.

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