Manifolds and Charts

Question

I have a very silly and basic question about finding charts for a manifold. The point is: I'm self learning differential geometry, however, I didn't find the answer for this in the book nor on the web. I've once asked about how to find charts, but now my point is another: is how to represent the elements of a manifold before start creating the charts.

First, I'm using Do Carmo's definition of manifold: A smooth manifold of dimension $n$ is a set $M$ with a family of bijective maps $\varphi_\alpha : U_\alpha \to M$ from open sets $U_\alpha\subset \mathbb{R}^n$ to $M$ such that:

1. $\bigcup_\alpha\varphi_\alpha(U_\alpha)=M$
2. For each pair $\alpha, \beta$ with $\varphi_\alpha(U_\alpha)\cap\varphi_\beta(U_\beta)=W\neq\emptyset$ we have $\varphi_\alpha^{-1}(W)$, $\varphi_\beta^{-1}(W)$ open in $\mathbb{R}^n$ and $\varphi_\beta^{-1}\circ\varphi_\alpha$, $\varphi_\alpha^{-1}\circ\varphi_\beta$ are differentiable.
3. The family $\left\{U_\alpha, \varphi_\alpha\right\}$ is maximum with respect to conditions 1 and 2.

Amongst all definitions of smooth manifold, this one was the one I prefered to work with. My point here is: we make this definition in order to avoid the need to consider manifolds as subsets of some euclidean space. In other words, we want to deal with them without making reference to some ambient space.

My problem with this is to find the charts. I have to construct bijective functions from $\mathbb{R}^n$ to $M$, and so my doubt is: how I describe the elements of $M$?

The classical example of finding charts for the sphere $S^n$ assumes that the sphere is defined as a subset of $\mathbb{R}^{n+1}$, so we know that if $p \in S^n$ then there are $n$ real numbers $p^i$ such that $p = \left(p^1 ,\cdots, p^n\right)$ that simply satisfy some conditions. Then it becomes easier to find the charts because first of all we know how to describe the elements of the set. Second, because we can use the ambient space, so we can use stereographic projections, which depends on the ambient space.

But in general, we don't want to use one ambient space. So, if I was asked for instance to find the charts for the sphere without the ambient space, what should I do? Well, now if $p \in S^n$, I cannot say that $p$ is one $n$-tuple of numbers, and I cannot also use things from outside $S^n$ like the planes and lines used in stereographic projections.

I'm confused with all of that. In understand the theorems, the proofs, the use of transition charts to ensure differentiability, and so on. My only problem is to find the charts, I feel I'm in need of examples, but I haven't found many. Do Carmo's examples deal just with surfaces, and he always presents them as subsets of $\mathbb{R}^3$.

Can someone explain those points or point me some references? Sorry for such a silly and basic question. And also sorry for the long text, I just didn't find a way to make it smaller.

Answer 1

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!

Answer 2

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.