Let \(U_i = \{[x]\in \mathbb{R}\mathbb{P}^n \mid x^i\neq 0\}\), and let \(\varphi_i\colon U_i\to \mathbb{R}^n\) be defined by

\[ \varphi_i[x^1,\ldots ,x^{n+1}] = \biggl(\frac{x^1}{x^i},\ldots , \frac{x^{i-1}}{x^i},\frac{x^{i+1}}{x^i},\ldots , \frac{x^{n+1}}{x^i}\biggr). \]

Then, \((U_i,\varphi_i)\) form an atlas on \(\mathbb{R}\mathbb{P}^n\).

Remark
  • The inverse of \(\varphi_i\) is given by \[ \varphi_i^{-1}(u^1,\ldots ,u^n) = [u^1,\ldots ,u^{i-1}, 1, u^{i+1}, \ldots , u^n]. \]
  • The transition maps are clearly smooth and thus \(\mathcal{A} = \{(U_i,\varphi_i), i=1,\ldots , n+1\}\) is a smooth atlas on \(\mathbb{R}\mathbb{P}^n\).