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

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(started on construction of rp^n)
 
(introduced the charts)
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This article contains information on possible {{link|chart||s}} for the real projective space of dimension <m>n</m>, denoted by <m>\RPn</m>.
 
This article contains information on possible {{link|chart||s}} for the real projective space of dimension <m>n</m>, denoted by <m>\RPn</m>.
 
We shall first define <m>\RPn</m>. Let <m>S^n = \left\{ (x_0, \dotsc, x_n) \middle\vert \sum_{i = 0}^n x_i^2 = 1 \right\}</m> be the <m>n</m>-sphere.
 
We shall first define <m>\RPn</m>. Let <m>S^n = \left\{ (x_0, \dotsc, x_n) \middle\vert \sum_{i = 0}^n x_i^2 = 1 \right\}</m> be the <m>n</m>-sphere.
Define a group action <m>{-1, 1} \cong \Ztwo</> on <m>S^n</m> by mapping <m>(\epsilon, x) \mapsto \epsilon x</m> with <m>epsilon \in {-1, 1}, x \in S^n</m>.
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Define a group action <m>\{-1, 1\} \cong \Ztwo</m> on <m>S^n</m> by mapping <m>(\epsilon, x) \mapsto \epsilon x</m> with <m>\epsilon \in \{-1, 1\}</m> and <m>x \in S^n</m>.
This group action is "nice enough" so that the quotient space <m>S^n / \Ztwo</m> is actually a real smooth compact Hausdorff manifold.
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This group action is "nice enough" so that the quotient space <m>S^n / \left( \Ztwo \right) </m> is actually a real smooth compact Hausdorff manifold.
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We now construct (the) (smooth) charts on {{M|\RPn}}.
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First we introduce some notation: if {{M|x \in \RPn}}, we write {{M|1=x = [x_0 : \dotsc : x_n]}} if {{M|(x_0, \dotsc, x_n)}} is a representative of the equivalence class {{M|x}}.
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Define the subsets {{M|U_i \subset \RPn}} for {{M|0 \leq i \leq n}} as
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{{MM|1=U_i := \{ [x_0 : \dotsc : x_n] \in \RPn \vert x_i \neq 0 \}. }}
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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 {{M|\R^n}} is chosen; see {{link|Real projective space}}).
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Now introduce maps
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<mm>
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\begin{align*}
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\phi_i: U_i & \to \R^n \\
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        [x_0 : \dotsc : x_{i - 1} : 1 : x_{i+1} : \dotsc : x_n] & \mapsto (x_0, \dotsc, \widehat{x_i}, \dotsc, x_n)
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\end{align*}
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</mm>
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where {{M| \widehat{x_i} }} denotes that the {{M|i}}-th coordinate is omitted.
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These maps are well-defined, and homeomorphisms if one takes the quotient topology on {{M|\RPn}}, and actually define a smooth structure on {{M|\RPn}}, as the transition maps {{M| \phi_j \circ \phi_i^{-1} }} are diffeomorphisms (where defined).

Revision as of 14:38, 19 February 2017

[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[/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.

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).