A geodesic polyarene in organic chemistry is a polycyclic aromatic hydrocarbon with curved convex or concave surfaces. Examples are fullerenes, nanotubes, corannulenes, helicenes and sumanene. The molecular orbitals of the carbon atoms in these systems are to some extent pyramidalized resulting a different pi electron density on either side of the molecule with consequences for activity.
One member of this group of organic compounds, pentaindenocorannulene (depicted below), can be considered a large fullerene fragment. The experimentally obtained curvature and degree of pyramidalizion (12.6° ) are both actually larger than that of fullerene but according to its discoverers the compound is relatively easy to synthesize starting from corannulene and a way is opened to produce larger such fragments by stitching.
Some bowl-shaped molecules reported in the literature are in fact partially hydrogenated. A C56H40 hydrocarbon has been synthesized containing 54 of the 60 carbon atoms found in fullerene 
- Geodesic polyarenes with exposed concave surfaces Lawrence T. Scott,† Hindy E. Bronstein, Dorin V. Preda, Ronald B. M. Ansems, Matthew S. Bratcher and Stefan Hagen Pure & Appl. Chem., Vol. 71, No. 2, pp. 209–219, 1999 Link
- Pentaindenocorannulene and Tetraindenocorannulene: New Aromatic Hydrocarbon Systems with Curvatures Surpassing That of C60Edward A. Jackson, Brian D. Steinberg, Mihail Bancu, Atsushi Wakamiya, and Lawrence T. Scott J. Am. Chem. Soc.; 2007; 129(3) pp 484 - 485; (Communication) doi:10.1021/ja067487h
- The two steps in this sequence are a Suzuki-Miyaura coupling with an ionic liquid and Tris(dibenzylideneacetone)dipalladium(0) and a Heck reaction with another palladium catalyst, DBU and DMAC
- by a so-called p-orbital axis vector (POAV). The value for fullerene is 11.6° and benzene 0°
- Synthesis of a C56H40 Hydrocarbon Bearing a 54-Carbon Framework of C60 Kung K. Wang, Yu-Hsuan Wang, Hua Yang, Novruz G. Akhmedov, and Jeffrey L. Petersen Org. Lett., 2009 doi:10.1021/ol900749a