A permanent magnet rotating electric machine comprises a stator having stator windings wound round a stator iron core and a permanent magnet rotor having a plurality of inserted permanent magnets in which the polarity is alternately arranged in the peripheral direction in the rotor iron core. The rotor iron core of the permanent magnets is composed of magnetic pole pieces, auxiliary magnetic poles, and a stator yoke, and furthermore has concavities formed on the air gap face of the magnetic pole pieces of the rotor iron core of the permanent magnets, gently tilting from the central part of the magnetic poles to the end thereof; resulting in a permanent magnet rotating electric machine effects of iron loss are reduced, and an electric car using highly efficient permanent magnet rotating electric machine are realized.