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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!

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    Also asked on MO: "Isometric embedding of a sphere," http://mathoverflow.net/questions/67139 .2011-06-08
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    yep indeed. I just realized stackexchange was more active that mathoverflow which is why I wanted to give a try here.2011-06-08
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    I posted a suggestion on MO, not a true answer, but ...2011-06-08
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    The overwhelming symmetry of the problem certainly suggests that the solution ought to be a sphere. It would be slightly a enlarged one, because the Euclidean distance underestimates geodesic distances between almost all pairs of points.2011-08-07
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    The radius of the sphere mentioned in Rahul Narain's comment would be $(6\pi-8)/9\doteq 1.20551$. The square root of the mean quadratic deviation would be $0.168261$, resulting mainly from pairs $(x,y)$ with large $\delta(x,y)$.2012-07-15

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