The vibronic (vibrational-electronic) interactions and the Jahn-Teller distortions in the mono- and trianions of corannulene, which shows a bowl-shaped C-5v structure in the neutral state, are discussed. Jahn-Teller active E-2 modes of vibration remove the degenerate E-1 state of the mono- and trianions to lead to C-s structures. We calculate the linear vibronic coupling constants in the corannulene anions using the B3LYP method, a hybrid (Hartree-Fock/density functional theory) method. The lowest two vibrational modes of 142 and 282 cm(-1) have large vibronic coupling constants, thus significantly contributing to the Jahn-Teller distortions. The electronic features of corannulene and coronene are discussed from a viewpoint that corannulene can be viewed as a fragment of C-60 and coronene as a fragment of graphite. Computed vibronic coupling constants in the corannulene anions are much larger than those in the coronene anions. The atomic motion in the Jahn-Teller active E-2 modes of corannulene is somewhat vertical to the bowl-shaped structure. On the other hand, the Jahn-Teller active E-2g modes of coronene and the resultant molecular distortions in its anions are strictly fixed on the molecular plane by symmetry. This contrast can explain the different vibronic features in corannulene and coronene. This fact may characterize the high-T-c superconductivity over 30 K in the alkali-metal complexes of C-60 and the low-T-c superconductivity in the graphite intercalation compounds.