  A class of graphene wound into three-dimensional periodic curved surfaces
("graphitic zeolites") is proposed and their electronic structures are obtained
to explore how the massless Dirac fermions behave on periodic surfaces. We find
in the tight-binding model that the low-energy band structure around the charge
neutrality point is dominated by the topology (cubic or gyroid) of the periodic
surface as well as by the spatial period $L$ in modulo 3 in units of the
lattice constant. In both of cubic and gyroid cases the Dirac electrons become
massive around the charge neutrality point, where the band gap is shown to
scale as $1/L$ within each mod-3 class. Wave functions around the gap are found
to have amplitudes sharply peaked around the topological defects that are
required to deform the graphene sheet into a three-dimensional periodic
surface, and this is shown to originate from non-trivial Bloch phases at $K$
and $K'$ points of the original graphene.
