Compact pseudo-Riemannian manifolds with parallel Weyl tensor

Details Compact pseudo-Riemannian manifolds with parallel Weyl tensor Compact pseudo-Riemannian manifolds with parallel Weyl tensor

2007-02-16

arxiv.org/abs/math/0702491v1

Annals of Global Analysis and Geometry vol. 37 (2010), no. 1, pp. 73-90

Mathematics

Differential Geometry

19 pages

Scientific paper

10.1007/s10455-009-9173-9

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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