  A family of connections on the space of couplings for a renormalizable field
theory is defined. The connections are obtained from a Levi-Civita connection,
for a metric which is a generalisation of the Zamolodchikov metric in two
dimensions, by adding a family of tensors which are solutions of the
renormalization group equation for the operator product expansion
co-efficients. The connections are torsion free, but not metric compatible in
general. The renormalization group flows of N=2 supersymmetric Yang-Mills
theory in four dimensions and the O(N)-model in three dimensions, in the large
$N$ limit, are analysed in terms of parallel transport under these connections.
