# Measure preserving coordinates of $S^2$ from $[0,1]^2$

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.

• – user1688 Oct 21 '18 at 17:42
• see gall-peters projection for a commonly used area preserving projection used in geographical maps. – Piyush Grover Oct 21 '18 at 17:44

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 $$\varphi$$.