Soliton geometry and the vacuum gravitational field equations
\textit{M. Crampin}, \textit{F. A. E. Pirani} and \textit{D. C. Robinson} [Lett. Math. Phys. 2, 15-19 (1977; Zbl 0363.35032)] gave a geometric interpretation of the inverse scattering method (ISM) in terms of soliton connections. Their work was based on flat connections i.e., connection 1- forms having zero curvature 2-forms. In this letter, the author introduces a new geometric setting (called ``soliton geometry'') which generalizes the approach by Crampin et. al. (loc. cit.) and extends the range of applicability of ISM, by using nonflat connections. Instead of imposing a zero curvature, it is required that the projection (which is a wedge product with the soldering form) of the curvature 2-form vanish. The flat soliton connection 1-form defined by Crampin et. al. (loc. cit.) is recovered when the soldering form is a 0-form; in this case the soliton geometry is the usual zero curvature formalism of ISM. The most important application of the introduced technique is connected with Einstein's vacuum field equations (for spacetime geometries possessing no symmetry) which are shown to admit a soliton connection and a soldering 1-form. For these field equations, the author has managed to construct the associated linear problem containing a spectral parameter.
scientific article; zbMATH DE number 4146712
scientific article