Roter: 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 ≥ 4 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 dis-crete parameters: the dimension n ≥ 4, the metric signature − −...++, and a sign factor ε = ±1 accounting for semidefiniteness of W, then the local-isometry types of our metrics g correspond bijectively to equivalence classes of surfaces Σ with equiaf-fine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface Σ 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 Σ, for a closed surface Σ 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