The Real Projective Space is an example of an abstract (smooth) manifold


For the sake of completeness, let us start with the definition of an abstract manifold.

Let \(X\) be a topological space. A chart (or coordinate chart) is an open set \(U\subseteq X\) together with a homeomorphism \(\phi: U \to \mathbb{R}^n\). An atlas is a collection of charts \( {(U_{\alpha},\phi_{\alpha})} \) with the condition \( X = \cup_{\alpha\in A} U_{\alpha} \). A smooth atlas is an atlas such that whenever \( U_{\alpha} \cap U_{\beta} \neq \emptyset \), the composition \( \phi_{\beta} \circ \phi_{\alpha}^{-1} : \phi_{\alpha}(U_{\alpha} \cap U_{\beta}) \to \phi_{\beta}(U_{\alpha}\cap U_{\beta}) \) i.e. every transition map is smooth. Two smooth atlases are equivalent if their union is also a smooth atlas. This gives a natural equivalence class of smoothly equivalent atlases which we call a smooth structure. More specifically, define an equivalence relation \( \sim \) where \( P \sim Q \) if \(P, Q\) are smooth atlases that are equivalent. The equivalence class generated by this relation is called a smooth structure.

A topological manifold \(X\) is a Hausdorff, second countable topological space \(X\) for which one can find an atlas. A smooth manifold \(X\) is a topological manifold with a distinguished smooth structure.

Before we talk about the real projective space \( \mathbb{R}P^n \), we talk about the \(n = 2\) case.

Definition (Real Projective Plane)

Let \(\mathbb{R}P^2\) denote the set of lines in \(\mathbb{R}^3\) that go through the origin. We say that \(\mathbb{R}P^2\) is the real projective plane.

To show this set is an abstract manifold, we first have to define a topology on it, which we do using the quotient topology. To do this consider \(\mathbb{R}^3\) and the equivalence relation \(\sim\) on it where \(x\sim y\) if and only if \(x = \lambda y\) for \(\lambda\in\mathbb{R},\lambda\neq 0.\) Now, consider the mapping \(p(x\in \mathbb{R}^3-{0}) = [x]\). Call \(Y\) the set mapped to by \(p\) and say that a set \(V\) in \(Y\) is open if and only if \(p^{-1}(V)\) is open in \(\mathbb{R}^3-0\). Then, \(p\) is clearly a quotient map so that \(Y\) is given the quotient topology. We can clearly identify \(\mathbb{R}P^2\) with \(Y\) by the mapping \(f: Y \to \mathbb{R}P^2\) where \(f(x) = L(x)\) where \(L(x)\) is the line that goes through the points \(x\) and \(0\). Via, this identification, we can say a set \(V\) in \(\mathbb{R}P^2\) is open if and only if \(f^{-1}(V)\) is open. Since \(Y\) and \(\mathbb{R}P^2\) are equivalent, we could have also defined \(\mathbb{R}P^2\) to simply be the topology given by quotient \((\mathbb{R}-0)/\sim\).

For each line, identify two antipodal points each magnitude 1 from the origin. This defines the set \(S^2\). A simple figure is provided below.

RP2

How does this fit in with my previous understanding of coordinates?

You may have thought coordinates as something which was implicitly connected to the space you were working in (e.g. \(\mathbb{R}, \mathbb{R}^2, …\)) in the way that giving a coordinate was the same as pointing out a unique place on the space. This is in fact, still correct. In this case, let \(X = \mathbb{R}^n\) and if our neighbourhood \(U \subset X\) is of dimension \(k\), then we let it be the identity mapping.