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Flat metrics with a prescribed derived coframing

Publication ,  Journal Article
Bryant, RL; Clelland, JN
Published in: Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
January 1, 2020

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.

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Published In

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

DOI

EISSN

1815-0659

Publication Date

January 1, 2020

Volume

16

Related Subject Headings

  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0105 Mathematical Physics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics
 

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Bryant, R. L., & Clelland, J. N. (2020). Flat metrics with a prescribed derived coframing. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 16. https://doi.org/10.3842/SIGMA.2020.004
Bryant, R. L., and J. N. Clelland. “Flat metrics with a prescribed derived coframing.” Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) 16 (January 1, 2020). https://doi.org/10.3842/SIGMA.2020.004.
Bryant RL, Clelland JN. Flat metrics with a prescribed derived coframing. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020 Jan 1;16.
Bryant, R. L., and J. N. Clelland. “Flat metrics with a prescribed derived coframing.” Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), vol. 16, Jan. 2020. Scopus, doi:10.3842/SIGMA.2020.004.
Bryant RL, Clelland JN. Flat metrics with a prescribed derived coframing. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2020 Jan 1;16.

Published In

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

DOI

EISSN

1815-0659

Publication Date

January 1, 2020

Volume

16

Related Subject Headings

  • 4904 Pure mathematics
  • 4901 Applied mathematics
  • 0105 Mathematical Physics
  • 0102 Applied Mathematics
  • 0101 Pure Mathematics