  We probe the effect of diffeomorphism symmetry on the critical exponents
values for massive O($N$) $\lambda\phi^{4}$ scalar field theories in curved
spacetime. We apply field-theoretic renormalization group tools, where we use
only momentum coordinates as opposed to the earlier standard procedure with
both space and momentum coordinates, for computing analytically the critical
exponents up to three-loop order for one of them and up to two-loop level for
the remaining ones. We found that the curved spacetime critical exponents are
the same as their flat spacetime counterparts, at least up to the level
considered. At the conclusions we present the physical interpretation for the
results.
