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Consider the unit sphere $S^2 = \left\{x\in\mathbb{R}^3 ~ {\large|} ~ |x|=1 \right\}$ and denote the uniform (Lebesgue) measures on the $S^2$ and $[0,1]^2$ by $m_S$ and $m_2$, respectively.

Question 1: Is there a map $\phi : [0,1]^2 \to S^2$ such that preserves the uniform measure, i.e., (with a slight abuse of notation) $\phi \circ m_2 = m_S \circ \phi$?

Question 2: Is there a way to construct coordinates on $(\varphi_1(w),\varphi_2(z)):[0,1]^2 \to S^2$ such that $dm_S(\varphi_1(w),\varphi_2(z)) = dw\,dz$?

Question 3: Does such coordinates exist for every higher dimensions?

Remark: On $S^1$, this is trivial by $\varphi (t) = e^{2\pi i t}$, up to a constant.

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For Question 1, see an earlier question. Both of the above measures can be "simulated" by measurable maps $f:[0,1]\to X$ where $X=[0,1]^2, S^2$, using the Lebesgue measure on the left. In fact, $X$ can be any compact metric space with any Borel probability measure.

This may not be the answer you want! Presumably, you are looking for continuous $\varphi$.

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