  Motivated by the problem of background independence of closed string field
theory we study geometry on the infinite vector bundle of local fields over the
space of conformal field theories (CFT's). With any connection we can associate
an excluded domain $D$ for the integral of marginal operators, and an operator
one-form $\omega_\mu$. The pair $(D, \omega_\mu)$ determines the covariant
derivative of any correlator of local fields. We obtain interesting classes of
connections in which $\omega_\mu$'s can be written in terms of CFT data. For
these connections we compute their curvatures in terms of four-point
correlators, $D$, and $\omega_\mu$. Among these connections three are of
particular interest. A flat, metric compatible connection $\HG$, and
connections $c$ and $\bar c$ having non-vanishing curvature, with $\bar c$
being metric compatible. The flat connection cannot be used to do parallel
transport over a finite distance. Parallel transport with either $c$ or $\bar
c$, however, allows us to construct a CFT in the state space of another CFT a
finite distance away. The construction is given in the form of perturbation
theory manifestly free of divergences.
