Hi,

Is there a way to find a function $F : \mathbb S^2 \rightarrow \mathbb R^3$ of class $\mathcal C^1$, minimizing 
$$\int\_{\mathbb S^2\times\mathbb S^2} (d(F(x),F(y)) - \delta(x,y))^2 ~dx ~dy$$
, where $d$ stands for the euclidean distance in $\mathbb R^3$ and $\delta$ the geodesic distance on the sphere $\mathbb S^2$? 

I tried to perform a Multi-Dimensional Scaling to get this least square solution for a finite set of point, but it seems that the solution was just the identity... is that right? 

Thanks!