Second-order symmetric Lorentzian manifolds: I. Characterization and general results

The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor—2–symmetric spacetimes—are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions. Related issues and applications are considered, and new open questions presented.