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2007 Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds
Andrzej Derdzinski, Witold Roter
Tohoku Math. J. (2) 59(4): 565-602 (2007). DOI: 10.2748/tmj/1199649875


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.


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Andrzej Derdzinski. Witold Roter. "Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds." Tohoku Math. J. (2) 59 (4) 565 - 602, 2007.


Published: 2007
First available in Project Euclid: 6 January 2008

zbMATH: 1146.53014
MathSciNet: MR2404206
Digital Object Identifier: 10.2748/tmj/1199649875

Primary: 53B30
Secondary: 58J99

Rights: Copyright © 2007 Tohoku University


Vol.59 • No. 4 • 2007
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