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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.

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A third degree surface and a touching sphere [closed]

Let consider a surface $z=1/(xy)$ and a sphere defined by $(x-1.5)^2+(y-1.5)^2+(z-1.5)^2=3/4$. The sphere touches the surface at (1,1,1). Is it possible to prove that point (1,1,1) is the only interse …