  The existence of an exactly marginal deformation in a conformal field theory
is very special, but it is not well understood how this is reflected in the
allowed dimensions and OPE coefficients of local operators. To shed light on
this question, we compute perturbative corrections to several observables in an
abstract CFT, starting with the beta function. This yields a sum rule that the
theory must obey in order to be part of a conformal manifold. The set of
constraints relating CFT data at different values of the coupling can in
principle be written as a dynamical system that allows one to flow arbitrarily
far. We begin the analysis of it by finding a simple form for the differential
equations when the spacetime and theory space are both one-dimensional. A
useful feature we can immediately observe is that our system makes it very
difficult for level crossing to occur.
