Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle.
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 discrete 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 equiaffine 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.
It is shown that in every dimension n = 3j + 2, j = 1, 2, 3,. . ., there exist compact pseudo-Riemannian manifolds with parallel Weyl tensor, which are Ricci-recurrent, but neither conformally flat nor locally symmetric, and represent all indefinite metric signatures. The manifolds in question are diffeomorphic to nontrivial torus bundles over the circle. They all arise from a construction that a priori yields bundles over the circle, having as the fibre either a torus, or a 2-step nilmanifold with a complete flat torsionfree connection; our argument only realizes the torus case.
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