Trilinear generally covariant equations of AP

Abstract

Field equations for n-frames h_a{}^μthat are possible in the theory of absolute parallelism (AP) are considered. The methods of compatibility (or formal integrability) theory enable us to find the non-Lagrangian equation having unusual kind of compatibility conditions, guaranteed by two (not one) identities. This 'unique equation' was not noted explicitly in the classification by Einstein and Mayer of compatible second order equations of AP. It is shown that some equations of AP (including 'unique equation') can be written in a trilinear form that contains only the matrix of frame density (of some weight) H_a{}^μand its derivatives and not inverse (coframe density) matrix. The equations are still regular and involutive for degenerate but finite matrices H_a{}^μif rank H_a{}^μ> 1.