We examine the electronic structure and total energies of schwarzites (negative-curvature analogs of the fullerenes). These structures have curvature induced via the presence of seven- or eight-membered rings. Ab initio dynamical relaxations reveal that these structures are closer in energy to graphite than are the fullerenes. Analysis of the electronic structure of these curved nets shows that they have a substantial gap of approximately 1.3 eV, which results from the curvature of the underlying structure.