Petrov types of slowly rotating fluid balls

Circularly rotating axisymmetric perfect fluid space-times are investigated to second order in the small angular velocity. The conditions of various s pecial Petrov types are solved in a comoving tetrad formalism. A number of theorems are stated on the possible Petrov types of various fluid models. I t is shown that Petrov type II solutions must reduce to the de Sitter space time in the static limit. Two space-times with a physically satisfactory en ergy-momentum tensor are investigated in detail. For the rotating incompres sible fluid, it is proven that the Petrov type cannot be D. The equation of the rotation function omega can be solved for the Tolman type IV fluid in terms of quadratures. It is also shown that the rotating version of the Tol man IV space-time cannot be Petrov type D.