  In this paper, I compare the generators of the renormalization group flow, or
the geometric $\beta$-functions for dimensional regularization and operator
regularization. I then extend the analysis to show that the geometric
$\beta$-function for a scalar field theory on a closed compact Riemannian
manifold is defined on the entire manifold. I then extend the analysis to find
the generator of the renormalization group flow for a conformal scalar-field
theories on the same manifolds. The geometric $\beta$-function in this case is
not defined.
