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]. 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</> on <m>S^n[/ilmath] by mapping [ilmath](\epsilon, x) \mapsto \epsilon x[/ilmath] with [ilmath]epsilon \in {-1, 1}, x \in S^n[/ilmath]. This group action is "nice enough" so that the quotient space [ilmath]S^n / \Ztwo[/ilmath] is actually a real smooth compact Hausdorff manifold.