Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds

Abstract

We determine the local structure of all pseudo-Riemannian manifolds $(M,g)$ in dimensions $n\ge4$ whose Weyl conformal tensor $W$ is parallel and has rank 1 when treated as an operator acting on exterior 2-forms at each point. If one fixes three discrete parameters: the dimension $n\ge4$, the metric signature $--...++$, and a sign factor $ε=\pm1$ accounting for semidefiniteness of $W$, then the local-isometry types of our metrics $g$ correspond bijectively to equivalence classes of surfaces $\varSigma$ with equiaffine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface $\varSigma$ arises, locally, as the leaf space of a codimension-two parallel distribution on $M$, naturally associated with $g$. We exhibit examples in which the leaves of the distribution form a fibration with the total space $M$ and base $\varSigma$, for a closed surface $\varSigma$ of any prescribed diffeomorphic type. Our result also completes a local classification of pseudo-Riemannian metrics with parallel Weyl tensor that are neither conformally flat nor locally symmetric: for those among such metrics which are not Ricci-recurrent, rank $W$ = 1, and so they belong to the class mentioned above; on the other hand, the Ricci-recurrent ones have already been classified by the second author.