Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, 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 $\mathbb S^2$?
Or $d$ could stand for the squared euclidean distance, and $\delta$ the square geodesic distance, if this makes the problem simpler. The goal is thus to approximate geodesic distances by euclidean distances of transformed points.
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 (or a uniform scaling)... is that right?
Thanks!