Using first-principles calculations we examine the band structures of ferromagnetic hexagonal manganites YXO3 (X=V, Cr, Mn, Fe, and Co) in the nonpolar nonsymmorphic P63/mmc space group. For YVO3 and YCrO3 we find a band inversion near the Fermi energy that generates a nodal ring in the kz=0 mirror plane. We perform a more detailed analysis for these compounds and predict the existence of the topological "drumhead" surface states. Finally, we briefly discuss the low-symmetry polar phases (space group P63cm) of these systems, and show they can undergo a P63/mmc→P63cm transition by condensation of soft K3 and Γ2- phonons. Based on our findings, stabilizing these compounds in the hexagonal phase could offer a promising platform for studying the interplay of topology and multiferroicity, and the coexistence of real-space and reciprocal-space topological protection in the same phase.