Flat Metrics with a Prescribed Derived Coframing

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $Ω= \big(Ω^1,Ω^2,Ω^3\big)$ of closed $2$-forms. One wants to construct a coframing $ω= \big(ω^1,ω^2,ω^3\big)$ of $M$ such that, first, ${\rm d}ω^i = Ω^i$ for $i=1,2,3$, and, second, the Riemannian metric $g=\big(ω^1\big)^2+\big(ω^2\big)^2+\big(ω^3\big)^2$ be flat. We show that, in the 'nonsingular case', i.e., when the three $2$-forms $Ω^i_p$ span at least a $2$-dimensional subspace of $Λ^2(T^*_pM)$ and are real-analytic in some $p$-centered coordinates, this problem is always solvable on a neighborhood of $p\in M$, with the general solution $ω$ depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution $ω$ can be taken to be a nonsingular cubic. Some singular situations are considered as well. In particular, we show that the problem is solvable locally when $Ω^1$, $Ω^2$, $Ω^3$ are scalar multiples of a single 2-form that do not vanish simultaneously and satisfy a nondegeneracy condition. We also show by example that solutions may fail to exist when these conditions are not satisfied.

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