Difference between revisions of "User:Harold/Charting RP^n"

Revision as of 16:44, 19 February 2017

[ilmath] \newcommand{\R}{\mathbb{R}} \newcommand{\RPn}{\R P^n} \newcommand{\Ztwo}{\mathbb{Z} / 2 \mathbb{Z}} \newcommand{\id}{\mathrm{Id}} [/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+1} [/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

We obtain the following transition maps:

• Let [ilmath] i,j \in\{1,\ldots,n\}\subseteq\mathbb{N} [/ilmath] be given.
  • In general the transition maps have the form: [ilmath] (\phi_j \circ \phi_i^{-1}) : \phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j)[/ilmath].
  • We obtain the transition map by case analysis, as follows:
    1. [ilmath]i\eq j[/ilmath] - in this case, the transition map is the identity map [ilmath] \underbrace{(\phi_i \circ \phi_i^{-1})}_{\eq \id_{\R^n} }: \underbrace{\phi_i(U_i \cap U_i)}_{\eq \R^n} \to \underbrace{\phi_i(U_i \cap U_i)}_{\eq \R^n} [/ilmath] given by [ilmath] \id_{\R^n} : x \mapsto x [/ilmath].
    2. [ilmath]i<j[/ilmath] - we obtain the following map:
       • [math] \phi_j \circ \phi_i^{-1} : \phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j)[/math] given by [math] (\phi_j \circ \phi_i^{-1})(x_0, \dotsc, x_{i-1}, x_{i+1}, \dotsc, x_n) \eq \frac{(x_0, \dotsc, x_{i - 1}, 1, x_{i+1}, \dotsc, x_{j-1}, x_{j+1}, \dotsc, x_n)}{x_j} [/math].
    3. [ilmath]i>j[/ilmath] - we obtain the following map:
       • [math] \phi_j \circ \phi_i^{-1} : \phi_i(U_i \cap U_j) \to \phi_j(U_i \cap U_j)[/math] given by [math] (\phi_j \circ \phi_i^{-1})(x_0, \dotsc, x_{i-1}, x_{i+1}, \dotsc, x_n) \eq \frac{(x_0, \dotsc, x_{j-1}, x_{j+1}, \dotsc, x_{i - 1}, 1, x_{i+1}, \dotsc, x_n)}{x_j} [/math].
  • This completes our case analysis
• Since [ilmath]i,j[/ilmath] were arbitrary we have shown this for all.
• These explicit expressions 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].