The electronic structures of 15 models of periodic graphitelike carbon surfaces (PGCS) with negative Gaussian curvatures are studied by a first-principles method. These models, which consist of P, D, and G surfaces of PGCS and range from 24 to 216 atoms in the unit cell, have different structural characteristics. Results on the band structures and density of states are presented. To achieve high accuracy, band-secular equations are solved at a large number of k points in the 1/48 of the Brillouin zone for each model. Of the 15 models studied, seven of them are insulating, seven are metallic, and one is a semimetal. Correlations with structural parameters of each model fail to pin down clearly the factor controlling the pp formation in PGCS, although it is found that the models containing only isolated seven- or eight-membered rings are insulating. Among all the models studied, the polybenzene model with only 24 atoms in a simple cubic unit cell is the most interesting-and has the largest indirect band gap of 2.96 eV.