Pseudo-Riemannian metric with neutral signature induced by solutions to Euler-Lagrange equations for a field of complex linear frames (Q362126)

Let \(G\) be a real semisimple Lie group of dimension \(n-1\) and consider the product manifold \(M=\mathbb R\times G\). The present paper is devoted to a \(\mathrm{GL}(n, \mathbb C)\)-invariant model for the field of complex linear frames \(E\) on \(M\) although using the complex field of frames instead of the real one did not reduce the computational difficulties resulting from the nonlinearity of field equations. NEWLINENEWLINENEWLINEAn important aspect of this study is that a solution \(E\) of the Euler-Lagrange equations yields a pseudo-Riemannian metric \(\gamma [E]\) on \(M\) and in the physical case of \(n=4\) there exist neutral such metrics \(\gamma [E]\). For this dimension special attention is paid to the case \(G=\mathrm{SL}(2, \mathbb{R})\).