  This is the first of three papers on the short-distance properties of the
energy-momentum tensor in field theory. We study the energy-momentum tensor for
renormalized field theory in curved space. We postulate an exact Ward identity
of the energy-momentum tensor. By studying the consistency of the Ward identity
with the renormalization group and diffeomorphisms, we determine the
short-distance singularities in the product of the energy-momentum tensor and
an arbitrary composite field in terms of a connection for the space of
composite fields over theory space. We discuss examples from the
four-dimensional $\phi^4$ theory. In the forthcoming two papers we plan to
discuss the torsion and curvature of the connection.
