Tohoku Mathematical Journal

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

Andrzej Derdzinski and Witold Roter

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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.

Article information

Tohoku Math. J. (2), Volume 59, Number 4 (2007), 565-602.

First available in Project Euclid: 6 January 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53B30: Lorentz metrics, indefinite metrics
Secondary: 58J99: None of the above, but in this section

Parallel Weyl tensor projectively flat connection null parallel distribution


Derdzinski, Andrzej; Roter, Witold. Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds. Tohoku Math. J. (2) 59 (2007), no. 4, 565--602. doi:10.2748/tmj/1199649875.

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