Exponential solutions of euler-lagrange equations for fields of complex linear frames on real space-time manifolds

We investigate a model of the field of complex linear frames on the product manifold M = ℝ × G, where G is a real semisimple Lie group. The model is invariant under the natural action of the group GL(n, ℂ) (n = dim M). It results in a modified Born-Infeld-type nonlinearity of field equations.We find a family of solutions of the Euler-Lagrange equations. These solutions are bases for the Lie algebra of left-invariant vector fields on ℝ × G “deformed” by a GL(n, ℂ)-valued mapping of the exponential form. Each solution induces a pseudo-Riemannian metric on M = ℝ × G. The normal-hyperbolic signature (in the physical case where n = 4) of this metric is not something aprioric and absolute, introduced “by hand” into our model but it is an intrinsic feature of solutions we found.