Flat Metrics with a Prescribed Derived Coframing

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Authors: Bryant, Robert L., Clelland, Jeanne N.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210606
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Flat Metrics with a Prescribed Derived Coframing. Robert L. Bryant and Jeanne N. Clelland. SIGMA 16 (2020), 004, 23 pages

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(T*ₚM) and are real-analytic in some p-centered coordinates, this problem is always solvable on a neighborhood of p∈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.