Spanish Relativity Meeting/ERE2009 Gravitation in the Large

Abstract

Abstract. We present the explicit local form of the metric of the second-order symmetric non-symmetric 4-dimensional Lorentzian manifolds. They turn out to be a specific subclass of plane waves. IntroductionThe historical roots of the subject under investigation go back to the classification of the locally symmetric spaces in the (proper) Riemannian case by Cartan [2,3]. Locally symmetric pseudoRiemannian manifolds are characterized by the condition ∇R = 0, where R denotes the curvature tensor of the manifold. The pseudo-Riemannian manifolds satisfying ∇ 2 R = 0 constitute a logical generalization of the locally symmetric ones. But in the Riemannian case these two conditions are equivalent. Moreover, in the Riemannian case: