We determine the local structure of all pseudo-Riemannian manifolds of dimensions
greater than 3 whose Weyl conformal tensor 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, the metric signature (with at least
two minuses and at least two pluses), and a sign factor accounting for
semidefiniteness of the Weyl tensor, then the local-isometry types of our
metrics correspond bijectively to equivalence classes of surfaces with
equiaffine projectively flat torsionfree connections; the latter equivalence
relation is provided by unimodular affine local diffeomorphisms. The surface
just mentioned arises, locally, as the leaf space of a codimension-two parallel
distribution on the pseudo-Riemannian manifold in question, naturally associated
with its metric. We construct examples showing that the leaves of this
distribution may form a fibration with the base which is a closed surface 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, the Weyl
tensor has rank 1, and so they belong to the class discussed in the previous
paragraph; on the other hand, the Ricci-recurrent ones have already been
classified by the second author.