# Flat metrics with a prescribed derived coframing

## Abstract

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $\Omega = (\Omega^1,\Omega^2,\Omega^3)$ of closed $2$-forms. One wants to construct a coframing $\omega= (\omega^1,\omega^2,\omega^3)$ of $M$ such that, first, $\mathrm{d}\omega^i = \Omega^i$ for $i=1,2,3$, and, second, the Riemannian metric $g=(\omega^1)^2+(\omega^2)^2+(\omega^3)^2$ be flat. We show that, in the `nonsingular case', i.e., when the three $2$-forms $\Omega^i_p$ span at least a $2$-dimensional subspace of $\Lambda^2(T^*_pM)$ 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 $\omega$ depending on three arbitrary functions of two variables. Moreover, the characteristic variety of the generic solution $\omega$ 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 $\Omega^1,\Omega^2,\Omega^3$ 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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### Scholars@Duke

#### Robert Bryant

My research concerns problems in the geometric theory of partial differential equations. More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy.

Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems. Because of their built-in invariance properties, these systems often have special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of *exterior differential systems*, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.

I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.

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