Manifolds and Charts

I hope that the following example is close to what you would like to see.

Let $ n \in \mathbb{N} $, and define an equivalence relation $ \sim $ on $ \mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \} $ as follows: $$ \forall \mathbf{x}_{1},\mathbf{x}_{2} \in \mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \}: \quad \mathbf{x}_{1} \sim \mathbf{x}_{2} \stackrel{\text{def}}{\iff} (\exists \lambda \in \mathbb{R} \setminus \{ 0 \}) (\mathbf{x}_{1} = \lambda \cdot \mathbf{x}_{2}). $$ Given any $ (x_{1},\ldots,x_{n+1}) \in \mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \} $, we denote its $ \sim $-equivalence class by $$ [x_{1}:\ldots:x_{n+1}]. $$ We call $ (\mathbb{R}^{n+1} \setminus \{ \mathbf{0}_{n+1} \})/\sim $ the real projective $ n $-space, and we usually denote it by $ \mathbb{R} \mathbb{P}^{n} $. Intuitively, one can think of $ \mathbb{R} \mathbb{P}^{n} $ as the set of straight lines in $ \mathbb{R}^{n+1} $ that pass through the origin.

Observe that $ \mathbb{R} \mathbb{P}^{n} $ was not born as a subset of some ambient Euclidean space $ \mathbb{R}^{N} $. Although the elements of $ \mathbb{R} \mathbb{P}^{n} $ can be visualized as straight lines in $ \mathbb{R}^{n+1} $, this visualization is irrelevant if we are to treat the elements as points of an abstract space. Hence, at the most fundamental level, we should view $ \mathbb{R} \mathbb{P}^{n} $ as simply a set equipped with an equivalence relation, without worrying over how it can be embedded into Euclidean space.

Despite the abstract nature of $ \mathbb{R} \mathbb{P}^{n} $, we can, curiously enough, endow it with a manifold structure. For each $ i \in \{ 1,\ldots,n + 1 \} $, define a subset $ U_{i} $ of $ \mathbb{R} \mathbb{P}^{n} $ as follows: $$ U_{i} := \{ [x_{1}:\ldots:x_{n+1}] ~|~ x_{i} \neq 0 \}. $$ Next, define ‘chart’ maps $ \varphi_{i}: U_{i} \to \mathbb{R}^{n} $ by $$ \varphi([x_{1}:\ldots:x_{n+1}]) \stackrel{\text{def}}{=} \left( \frac{x_{0}}{x_{i}},\ldots,\widehat{\frac{x_{i}}{x_{i}}},\ldots,\frac{x_{n+1}}{x_{i}} \right) \in \mathbb{R}^{n}, $$ where the $ ~ \widehat{\hspace{4mm}} ~ $-symbol indicates an omitted term. Then $ \{ (U_{i},\varphi_{i}) \}_{i=1}^{n+1} $ is an atlas that makes $ \mathbb{R} \mathbb{P}^{n} $ an $ n $-dimensional manifold. We shall leave the derivation of the transition maps as an exercise.

In each case, constructing charts from first principles requires usually some ingenuity. This is why differential geometry in Euclidean space is so much easier-the space comes equipped with very natural charts(i.e. Cartesian,plane and cylindrical polar coordinates,spherical coordinates). You're in luck since differential geometry of all the mathematical disciplines, has the largest number of clear textbooks for self learning. Not only are there a ton of good actual textbooks, there's a wealth of online resources, like Nigel Hitchen's wonderful lecture notes on manifolds. The best places to start seriously learning by self-study about manifolds with lots of examples are probably the textbooks by John M.Lee, Introduction To Topological Manifolds and Introduction to Smooth Manifolds, both in thier second editions. Both have lots of examples and wonderful pictures. Another, much cheaper book that I think you'll find helpful is David Gauld's Differential Topology:An Introduction, now in Dover. It uses some unusual concepts of nearness to define basic topology, but I'd just ignore these if you already know basic topology since they're equivalent to the usual. The good part of Gauld is that it has very detailed examples of charts in Euclidean spaces and thier related embedding theorums. This will help you understand how charts are constructed on abstract manifolds. You also should take a look at Loring Tu's An Introduction to Manifolds. It has a very clear,beautiful and visual presentation of the material that's shorter and requires only undergraduate analysis and algebra to understand. And finally, one last book you'll find useful is An Introduction To Differential Manifolds And Riemannian Geometry by William M. Boothby, which gives a wonderful bridge course between "advanced calculus" and a modern course on differentiable manifolds, complete with many concrete computations and examples of charts on both Euclidean and abstract manifolds. I think you'll particularly find Boothby helpful to clear up a lot of your questions. Good luck!