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Imagine a sphere of liquid in $\mathbb{R}^3$. The sphere is rotating around a vector with some coordinates $(1,0,0)$ and reached a hydrostatic equilibrium, so it becomes an ellipsoid. What would happen if it was rotating around a vector with coordinates $(i,0,0)$?

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Well $(i,0,0)$ does not belong to $\mathbb R^3$... –  Mariano Suárez-Alvarez Feb 7 '11 at 5:03
Yes, I just guess what would happen if we change to $\mathbb{C}^3$ –  Anixx Feb 7 '11 at 5:08
I agree with Mariano's comment: at present the question seems simply counterfactual. There are of course analogous things for $\mathbb{C}$-vector spaces to the group $\operatorname{SO}(n)$ of rotations of a finite-dimensional real inner product space, but in order to get a real [sic!] question you need to specify exactly what you have in mind. (Or possibly specify a real-world problem for which we can attempt to choose a reasonable mathematical model, but I don't know what a "complex liquid" is supposed to be...) –  Pete L. Clark Feb 7 '11 at 5:09

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