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Given a 2D manifold endowed with the metric of a sphere $S_2$, how many ways can it be analytically embedded into 3D space?

By an analytic embedding we mean that the parametric equation of the surface can be written in terms of analytic functions.

(This is different from a $C^\infty$ embedding.)

To put it another way, is it possible to crumple a sphere without giving it any hard creases?

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First, you need a complex structure. And then (quote from the link):

Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds.

For example, the Whitney embedding theorem tells us that every smooth $n$-dimensional manifold can be embedded as a smooth submanifold of $\Bbb R ^{2n}$, whereas it is "rare" for a complex manifold to have a holomorphic embedding into $\Bbb C^n$. Consider for example any compact connected complex manifold $M$: any holomorphic function on it is constant by Liouville's theorem. Now if we had a holomorphic embedding of $M$ into $\Bbb C^n$, then the coordinate functions of $\Bbb C^n$ would restrict to nonconstant holomorphic functions on $M$, contradicting compactness, except in the case that $M$ is just a point.

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