The morphology of fullerene networks can be widely extended by introducing heptagonal or octagonal rings, which produce a Gaussian negative curvature. Their presence makes it possible to form donut-, coil- and sponge-shaped networks of carbon atoms. We discuss the geometry of the polymorphous forms based on the net diagram method relative to a honeycomb lattice, and further study the electronic structures constructed by the network of π electrons system. Special emphasis is put on how the geometrical parameters, which specify the relative arrangement of polygonal rings, control the electronic structures in the various extended-fullerene networks. In addition, we mention that the presence of a certain type of edge in fullerene network derives critical localized edge states at the Fermi level.