  We present two families of exterior differential systems (EDS) for
non-isometric embeddings of orthonormal frame bundles over Riemannian spaces of
dimension q = 2, 3, 4, 5.... into orthonormal frame bundles over flat spaces of
sufficiently higher dimension. We have calculated Cartan characters showing
that these EDS satisfy Cartan's test and are well-posed dynamical field
theories. The first family includes a constant-coefficient (cc) EDS for
classical Einstein vacuum relativity (q = 4). The second family is generated
only by cc 2-forms, so these are integrable (but nonlinear) systems of partial
differential equations. These calibrated field theories apparently are new,
although the simplest case q = 2 turns out to embed a ruled surface of
signature (1,1) in flat space of signature (2,1). Cartan forms are found to
give explicit variational principles for all these dynamical theories.
