  Discussed are field-theoretic models with degrees of freedom described by the
$n$-leg field in an $n$-dimensional "space-time" manifold. Lagrangians are
generally-covariant and invariant under the internal group GL$(n,{\bf R})$. It
is shown that the resulting field equations have some correspondence with
Einstein theory and possess homogeneous vacuum solutions given by semisimple
Lie group spaces or their appropriate deformations. There exists a
characteristic link with the generalized Born-Infeld type nonlinearity and
relativistic mechanics of structured continua. In our model signature is not
introduced by hands, but is given by integration constants for certain
differential equations.
