  In this paper we investigate analytically the conformal symmetry influence on
the next-to-leading order radiative quantum corrections to critical exponents
for massless O($N$) $\lambda\phi^{4}$ scalar field theories in curved
spacetime. We renormalize the theory by applying the BPHZ method. We find that
the critical exponents are the same as that of flat spacetime, at least at the
loop order considered. We argue that this result agrees perfectly with the
universality hypothesis.
