Soliton geometry and the vacuum gravitational field equations
type: Item
instance of: scholarly article
title: Soliton geometry and the vacuum gravitational field equations
zbMATH Open document ID: 0699.35240
DOI: 10.1007/BF00401589
author: Metin Gürses
published in: Letters in Mathematical Physics
publication date: 1989-01-01T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 35Q15
Mathematics Subject Classification ID: 83C05
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35N10
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 4146712
zbMATH Keywords: Lax pairs
zbMATH Keywords: inverse scattering method
zbMATH Keywords: soliton connections
zbMATH Keywords: zero curvature
reviewed by: I. V. Barashenkov
MaRDI profile type: Publication
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cites work: Cartan ideal, prolongation, and Bäcklund transformations for Einstein’s equations
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description: scientific article; zbMATH DE number 4146712