  We characterize the conjugate linearized Ricci flow and the associated
backward heat kernel on closed three--manifolds of bounded geometry. We discuss
their properties, and introduce the notion of Ricci flow conjugated constraint
sets which characterizes a way of Ricci flow averaging metric dependent
geometrical data. We also provide an integral representation of the Ricci flow
metric itself and of its Ricci tensor in terms of the heat kernel of the
conjugate linearized Ricci flow. These results, which readily extend to closed
n-dimensional manifolds, yield for various conservation laws, monotonicity and
asymptotic formulas for the Ricci flow and its linearization.
