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Given the function $x:[0,\pi]\times[0,\pi]\times[0, 2\pi]\rightarrow \mathbb{R}^4$ where $x(\theta_1,\theta_2,\varphi)=(\cos\theta_1,\sin\theta_1 \cos\theta_2, sin\theta_1 \sin\theta_2 \cos\varphi, \sin\theta_1 \sin\theta_2 \sin\varphi)$, I want to show that $$\text{Im}(x)=S^3$$ where $S^3$ is the 4-dimensional unit sphere.

I have shown that $\text{Im}(x)\subseteq S^3$. But now I have problems showing the other direction.

Could someone please help me?

Thank you!

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Let $(w,x,y,z)$ be a coordinate on $S^3$. Break this quadruple into two bits, $w$ and $(x,y,z)$. Let $l_1$ be the length of the first of those, and $l_2$ be the length of the second. Then $\sqrt{l_1^2 + l_2^2}=1$. Therefore, set $\cos\theta_1$ equal to $l_1$ and $\sin\theta_1$ equal to $l_2$. Then divide all of $x$, $y$ and $z$ in $(x,y,z)$ by $\sin\theta_1$ and repeat the process.

Formally, this should be provable by induction.

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  • $\begingroup$ Thank you very much! You helped a lot. $\endgroup$ – TwoStones Apr 8 at 18:30

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