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Suppose I have a two-manifold homeomorphic to the disk, and I prescribe a first and second fundamental form $a$ and $b$, and that these forms are compatible, in the sense of satisfying the Gauss-Codazzi-Mainardi equations.

It is well-known that the manifold may not be isometrically immersible in $\mathbb{R}^3$ for any $b$. So let's suppose also that the manifold has at least one isometric immersion. Must one of these immersions have second fundamental form $b$?

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    I'm confused. In the case you cite, there is no $b$ satisfying the integrability conditions. If you in fact have an $a$ and $b$ satisfying the compatibility conditions, then you're golden. So is your question really just for what $a$ there exists an isometric immersion? – 2017-01-06
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    @TedShifrin I give you two tensor fields $a$, $b$ on an abstract two-manifold, and these tensor fields satisfy the compatibility equations. Does there then exist an immersion of the manifold whose first fundamental form is $a$ and whose second is $b$? Or are you saying that the compatibility conditions are enough to guarantee I can "integrate up" the immersion given the tensor fields? – 2017-01-06
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    Yes, I'm saying that the Fundamental Theorem of Surface Theory (proved first by Bonnet) says precisely that. – 2017-01-06
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    @TedShifrin Great, thanks. If you post an answer with a reference, I'll upvote and accept it (otherwise I'll answer myself). – 2017-01-07

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The Fundamental Theorem of Surface Theory says precisely that, given symmetric bilinear forms $a$ and $b$, the Gauss and Mainardi-Codazzi equations are precisely the integrability conditions to obtain an immersed surface in $\Bbb R^3$ (on a simply connected domain). In fact, it holds more generally for hypersurfaces in $\Bbb R^{n+1}$.

See, for example, Spivak's A Comprehensive Introduction to Differential Geometry, Volume 3, pp. 56-59, or Kobayashi-Nomizu, Foundations of Differential Geometry, Volume 2, pp. 47-51. One can also prove it, in differential forms version, as an application of the Frobenius Theorem. For this, see Chern, Chen, Lam, Lectures on Differential Geometry, Section 6.3.