Sphere (topological manifold)

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Would be good example

Definition

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

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

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

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

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

Proof of claims

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

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