Some Applications of the Infinitesimal-Holonomy Group to the Petrov Classification of Einstein Spaces

The classifications of Einstein spaces by Schell and Petrov are combined and certain nonlocal results are obtained. In particular, we show that an Einstein space cannot be type I with a rank four Riemann tensor in a four-dimensional region. On using the notion of a perfect or imperfect infinitesimal-holonomy group, we establish the conditions under which an Einstein space possesses a two-, four-, or six-parameter group. We find that two- and four-parameter groups are associated with special cases of type II null and type III, respectively.