User:Harold/Charting RP^n

$\newcommand{\R}{\mathbb{R}} \newcommand{\RPn}{\R P^n} \newcommand{\Ztwo}{\mathbb{Z} / 2 \mathbb{Z}}$ This article contains information on possible charts for the real projective space of dimension $n$, denoted by $\RPn$. We shall first define $\RPn$. Let $S^n = \left\{ (x_0, \dotsc, x_n) \middle\vert \sum_{i = 0}^n x_i^2 = 1 \right\}$ be the $n$-sphere. Define a group action ${-1, 1} \cong \Ztwo</> on <m>S^n$ by mapping $(\epsilon, x) \mapsto \epsilon x$ with $epsilon \in {-1, 1}, x \in S^n$. This group action is "nice enough" so that the quotient space $S^n / \Ztwo$ is actually a real smooth compact Hausdorff manifold.