Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 004, 23 pages      arXiv:1908.01041

Flat Metrics with a Prescribed Derived Coframing

Robert L. Bryant a and Jeanne N. Clelland b
a) Duke University, Mathematics Department, P.O. Box 90320, Durham, NC 27708-0320, USA
b) Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA

Received August 28, 2019, in final form January 09, 2020; Published online January 20, 2020

The following problem is addressed: A $3$-manifold $M$ is endowed with a triple $\Omega = \big(\Omega^1,\Omega^2,\Omega^3\big)$ of closed $2$-forms. One wants to construct a coframing $\omega = \big(\omega^1,\omega^2,\omega^3\big)$ of $M$ such that, first, ${\rm d}\omega^i = \Omega^i$ for $i=1,2,3$, and, second, the Riemannian metric $g=\big(\omega^1\big)^2+\big(\omega^2\big)^2+\big(\omega^3\big)^2$ 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.

Key words:exterior differential systems; metrization.

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