On four-dimensional curvature recurrent spaces with a neutral metric

This paper discusses recurrence features of the Riemann curvature tensor and its relationship with the Weyl conformal tensor. We classify the $4$-dimensional recurrent spaces equipped with a neutral metric according to the algebraic types of these tensor fields. We use the holonomy theory and the eigenbivector structure of the curvature tensor for this metric signature. In particular, we determine all possible holonomy types for $4$-dimensional spaces of recurrent curvature. In this analysis, the structures of recurrent and parallel vector fields and bivectors associated with the holonomy type considered are also useful. We finally give some relevant remarks on the Weyl conformal tensor, and present several examples related to the study.