  The presence of topological edge modes at the interface of two perturbed
honeycomb photonic crystals with $C_6$ symmetry is often attributed to the
different signs of Berry curvature at the K and K$'$ valleys. In contrast to
the electronic counterpart, the Chern number defined in photonic valley Hall
effect is not a quantized quantity but can be tuned to finite values including
zero simply by changing geometrical perturbations. Here, we argue that the edge
modes in photonic valley Hall effect can exist even when Berry curvature
vanishes. We numerically demonstrate the presence of the zero-Berry-curvature
edge modes in triangular lattice photonic crystal slab structures in which
$C_3$ symmetry is maintained but inversion symmetry is broken. We investigate
the evolution of the Berry curvature from the honeycomb-lattice photonic
crystal slab to the triangular-lattice photonic crystal slab and show that the
triangular-lattice photonic crystals still support edge modes in a very wide
photonic bandgap. Additionally, we find that the edge modes with zero Berry
curvature can propagate with extremely low bending loss.
