Sphere (topological manifold)

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Would be good example

Definition

Let [ilmath]\mathbb{S}^n\subseteq\mathbb{R}^{n+1} [/ilmath] denote the usual [ilmath]n[/ilmath]-sphere, of course defined by:

• [ilmath]\mathbb{S}^n:\eq\{x\in\mathbb{R}^{n+1}\ \big\vert\ \Vert x\Vert\eq 1 \} [/ilmath] [ilmath]:\eq\left\{(x_1,\ldots,x_{n+1})\in\mathbb{R}^{n+1}\ \left\vert\ \sqrt{\sum^{n+1}_{i\eq 1}(x_i^2)\eq 1}\right.\right\} [/ilmath] - where [ilmath]\Vert\cdot\Vert[/ilmath] is the Euclidean norm on Euclidean [ilmath](n+1)[/ilmath]-space

Note that in this article [ilmath]\mathbb{B}^n\subseteq\mathbb{R}^n[/ilmath] with [ilmath]\mathbb{B}^n:\eq\{x\in\mathbb{R}^n\ \big\vert\ \Vert x\Vert < 1\} [/ilmath] is the open unit ball (with centre at the origin, as is implied by the name)

We claim that [ilmath]\mathbb{S}^n[/ilmath] is a topological manifold with the following standard [ilmath]2n+2[/ilmath] charts:

• For [ilmath]i\in\{1,\ldots,n+1\}\subseteq\mathbb{N} [/ilmath]:
  • Define [ilmath]U^+_i:\eq\{(x_1,\ldots,x_{n+1})\in\mathbb{S}^n\ \big\vert\ x_i>0\} [/ilmath]
    • [ilmath]\varphi^+_i:U^+_i\rightarrow\mathbb{B}^n[/ilmath] given by [ilmath]\varphi^+_i:(x_1,\ldots,x_{n+1})\mapsto(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})[/ilmath]
      • So that [ilmath](U^+_i,\varphi^+_i)[/ilmath] is a chart
  • Define [ilmath]U^-_i:\eq\{(x_1,\ldots,x_{n+1})\in\mathbb{S}^n\ \big\vert\ x_i<0\} [/ilmath]
    • [ilmath]\varphi^-_i:U^-_i\rightarrow\mathbb{B}^n[/ilmath] given by [ilmath]\varphi^-_i:(x_1,\ldots,x_{n+1})\mapsto(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})[/ilmath]
      • So that [ilmath](U^-_i,\varphi^-_i)[/ilmath] is a chart

Proof of claims

• Note that Hausdorff is inherited from [ilmath]\mathbb{R}^{n+1} [/ilmath]
  • As is [ilmath]\mathbb{S}^{n} [/ilmath] being a second countable topological space
• The crux lies in this locally euclidean part.
  • Define: [ilmath]f:\mathbb{B}^n\rightarrow\mathbb{R} [/ilmath] given by [ilmath]f:x\mapsto\sqrt{1-\Vert x\Vert ^2} [/ilmath]
    • We note that:
      1. [ilmath]U^+_i[/ilmath] is the graph of [ilmath]x_i\eq f(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})[/ilmath], and
      2. [ilmath]U^-_i[/ilmath] is the graph of [ilmath]x_i\eq-f(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_{n+1})[/ilmath]

Grade: A

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Would be good to do Alec (talk) 05:05, 12 March 2017 (UTC)