Flat Metrics with a Prescribed Derived Coframing

Robert L. Bryant ; Jeanne N. Clelland

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

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.

Détail

Keywords: exterior differential systems, metrization.

Bibliographie

[1] Acharya A., “Stress of a spatially uniform dislocation density field”, J. Elasticity, 137 (2019), 151–155, arXiv: 1809.03567 | DOI | MR | Zbl

[2] Bryant R. L., Chern S. S., Gardner R. B., Goldschmidt H. L., Griffiths P. A., Exterior differential systems, Mathematical Sciences Research Institute Publications, 18, Springer-Verlag, New York, 1991 | DOI | MR | Zbl

[3] Ivey T. A., Landsberg J. M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, 175, 2nd ed., Amer. Math. Soc., Providence, RI, 2016 | DOI | MR | Zbl