  We construct invariant differential operators acting on sections of vector
bundles of densities over a smooth manifold without using a Riemannian metric.
The spectral invariants of such operators are invariant under both the
diffeomorphisms and the gauge transformations and can be used to induce a new
theory of gravitation. It can be viewed as a matrix generalization of Einstein
general relativity that reproduces the standard Einstein theory in the weak
deformation limit. Relations with various mathematical constructions such as
Finsler geometry and Hodge-de Rham theory are discussed.
