Renormalization group (RG) flows in theory space (the space of couplings) a
re generated by a vector field-the beta function. Using a specific metric a
nsatz in theory space and certain methods employed largely in the context o
f general relativity, we examine the nature of the expansion, shear and rot
ation of geodesic RG flows. The expansion turns out to be a negative quanti
ty inversely related to the norm of the beta function. This implies the foc
using of the flows towards the fixed points of a given field theory. The ev
olution equation for the expansion along geodesic RG flows is written down
and analyzed. We illustrate the results for a scalar field theory with a j
phi coupling and pointers to other areas are briefly mentioned.