Using rings of carbon with five atoms, closed structures like fullerenes with positive Gaussian curvature can be built. If we introduce rings with more than six atoms, periodic negatively curved graphite can be constructed. Starting from the Weierstrass representation we have generated graphite in which the atoms lie in the exact D, G and P triply periodic minimal surfaces. These structures divide space into two congruent labyrinths in which the inside is the same as the outside. We also have found that using pentagons and octagons, a new periodic graphitic sheet with the same topology as the I-WP minimal surface can be obtained. Densities, curvatures and cell parameters are given. The geometry and topology of curved graphitic structures including fullerenss is explained. Finally, different mathematical transformations which can give an insight into kinematic processes of curved graphite are analyzed. Here, we find that a variety of new shapes are possible.