User:Harold/Charting RP^n
[ilmath] \newcommand{\R}{\mathbb{R}} \newcommand{\RPn}{\R P^n} \newcommand{\Ztwo}{\mathbb{Z} / 2 \mathbb{Z}} [/ilmath] This article contains information on possible charts for the real projective space of dimension [ilmath]n[/ilmath], denoted by [ilmath]\RPn[/ilmath].
Definition of [ilmath]\RPn[/ilmath]
We shall first define [ilmath]\RPn[/ilmath]. Let [ilmath]S^n = \left\{ (x_0, \dotsc, x_n) \middle\vert \sum_{i = 0}^n x_i^2 = 1 \right\}[/ilmath] be the [ilmath]n[/ilmath]-sphere. Define a group action [ilmath]\{-1, 1\} \cong \Ztwo[/ilmath] on [ilmath]S^n[/ilmath] by mapping [ilmath](\epsilon, x) \mapsto \epsilon x[/ilmath] with [ilmath]\epsilon \in \{-1, 1\}[/ilmath] and [ilmath]x \in S^n[/ilmath]. This group action is "nice enough" so that the quotient space [ilmath]S^n / \left( \Ztwo \right) [/ilmath] is actually a real smooth compact Hausdorff manifold.
Construction of the charts
We now construct (the) (smooth) charts on [ilmath]\RPn[/ilmath]. First we introduce some notation: if [ilmath]x \in \RPn[/ilmath], we write [ilmath]x = [x_0 : \dotsc : x_n][/ilmath] if [ilmath](x_0, \dotsc, x_n)[/ilmath] is a representative of the equivalence class [ilmath]x[/ilmath]. Define the subsets [ilmath]U_i \subset \RPn[/ilmath] for [ilmath]0 \leq i \leq n[/ilmath] as [math]U_i := \{ [x_0 : \dotsc : x_n] \in \RPn \vert x_i \neq 0 \}.[/math] This is well-defined, because the choice of representative only depends on a sign or a non-zero scalar multiple (if the definition of lines in [ilmath]\R^n[/ilmath] is chosen; see Real projective space). Now introduce maps
[math] \begin{align*} \phi_i: U_i & \to \R^n \\ [x_0 : \dotsc : x_{i - 1} : 1 : x_{i+1} : \dotsc : x_n] & \mapsto (x_0, \dotsc, \widehat{x_i}, \dotsc, x_n) \end{align*} [/math]
where [ilmath] \widehat{x_i} [/ilmath] denotes that the [ilmath]i[/ilmath]-th coordinate is omitted. These maps are well-defined, and homeomorphisms if one takes the quotient topology on [ilmath]\RPn[/ilmath], and actually define a smooth structure on [ilmath]\RPn[/ilmath], as the transition maps [ilmath] \phi_j \circ \phi_i^{-1} [/ilmath] are diffeomorphisms (where defined).
On the transition maps
The transition maps [ilmath] \phi_j \circ \phi_i^{-1} [/ilmath] are defined on [ilmath] \phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j)[/ilmath]. They are explicitly given by mapping [ilmath](x_0, \dotsc, x_{i-1}, x_{i+1}, \dotsc, x_n)[/ilmath] to [ilmath][x_0 : \dotsc : x_{i - 1} : 1 : x_{i+1} : \dotsc : x_n][/ilmath] under [ilmath] \phi^{-1} [/ilmath], and then mapping [ilmath][x_0 : \dotsc : x_{i - 1} : 1 : x_{i+1} : \dotsc : x_n] \eq \left[\frac{x_0}{x_j} : \dotsc : \frac{x_{i - 1} }{x_j} : \frac{1}{x_j} : \frac{x_{i+1} }{x_j} : \dotsc : \frac{x_n}{x_j} \right] [/ilmath] to [ilmath] \frac{(x_0, \dotsc, x_{i - 1}, 1, x_{i+1}, \dotsc, x_{j-1}, x_{j+1}, \dotsc, x_n)}{x_j} [/ilmath]. This explicit expression makes it obvious that the transition maps are smooth, and they are obviously invertible, with smooth inverse. As such, they are diffeomorphisms from [ilmath]\phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j)[/ilmath].
