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

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Abstract
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.

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