Tohoku Mathematical Journal
2007
December
SECOND SERIES VOL. 59, NO. 4

Tohoku Math. J.
59 (2007), 565-602

Title PROJECTIVELY FLAT SURFACES, NULL PARALLEL DISTRIBUTIONS, AND CONFORMALLY SYMMETRIC MANIFOLDS

Author Andrzej Derdzinski and Witold Roter

(Received May 2, 2006, revised September 25, 2006)
Abstract. 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.

2000 Mathematics Subject Classification. Primary 53B30; Secondary 58J99.
Key words and phrases. Parallel Weyl tensor, projectively flat connection, null parallel distribution.

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