# Flat metrics with a prescribed derived coframing

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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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https://hdl.handle.net/10161/19200Collections

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Show full item record### Scholars@Duke

#### Robert Bryant

Phillip Griffiths Professor of Mathematics

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 specia

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