Flat metrics with a prescribed derived coframing

The following problem is addressed: A 3-manifold M is endowed with a triple Ω =(Ω¹, Ω², Ω³) of closed 2-forms. One wants to construct a coframing ω =(ω¹, ω², ω³) of M such that, first, dωⁱ = Ωⁱ for i = 1, 2, 3, and, second, the Riemannian metric g = (ω¹)² + (ω²)² + (ω³)² be flat. We show that, in the ‘nonsingular case’, i.e., when the three 2-forms Ωⁱp span at least a 2-dimensional subspace of Λ²(Tp*M) and are real-analytic in some p-centered coordinates, this problem is always solvable on a neighborhood of p (Formula Presented) 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 Ω¹, Ω², Ω³ 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.

Duke Scholars

Author Robert Bryant Mathematics