  Based on the principle of relativity, we find that the sufficient and
necessary condition for the general covariance of a field theory actually
requires more than the invariance of its local Langrangian density. If the
spacetime is not a flat one, its derivative requirement from the analysis of
the parallel transportation of tensor fields over spacetime restricts the
generally supposed covariance group, the group of differmorphisms, to the group
of linear coordinate transformations. Moreover, for any a field theory with
linear equations of motion, it stipulates for a universal physical propagation
speed of interaction over the spacetime manifold.
