  An analysis of one and two point functions of the energy momentum tensor on
homogeneous spaces of constant curvature is undertaken. The possibility of
proving a $c$-theorem in this framework is discussed, in particular in relation
to the coefficients $c,a$, which appear in the energy momentum tensor trace on
general curved backgrounds in four dimensions. Ward identities relating the
correlation functions are derived and explicit expressions are obtained for
free scalar, spinor field theories in general dimensions and also free vector
fields in dimension four. A natural geometric formalism which is independent of
any choice of coordinates is used and the role of conformal symmetries on such
constant curvature spaces is analysed. The results are shown to be constrained
by the operator product expansion. For negative curvature the spectral
representation, involving unitary positive energy representations of
$O(d-1,2)$, for two point functions of vector currents is derived in detail and
extended to the energy momentum tensor by analogy. It is demonstrated that, at
non coincident points, the two point functions are not related to $a$ in any
direct fashion and there is no straightforward demonstration obtainable in this
framework of irreversibility under renormalisation group flow of any function
of the couplings for four dimensional field theories which reduces to $a$ at
fixed points.
