New examples of compact Weyl-parallel manifolds

TL;DR

This paper constructs new examples of compact pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric, covering all indefinite signatures in dimensions five and higher.

Contribution

It demonstrates the existence of such manifolds in all indefinite signatures and dimensions ≥5, extending previous knowledge limited to specific dimensions.

Findings

Existence of compact pseudo-Riemannian manifolds with parallel Weyl tensor in all signatures

Construction of examples in all dimensions n ≥ 5

Manifolds are diffeomorphic to nontrivial torus bundles over the circle

Abstract

We prove the existence of compact pseudo-Riemannian manifolds with parallel Weyl tensor which are neither conformally flat nor locally symmetric, and represent all indefinite metric signatures in all dimensions $\,n\ge5$. Until now such manifolds were only known to exist in dimensions $\,n=3j+2$, where $\,j\,$ is any positive integer [11]. As in [11], our examples are diffeomorphic to nontrivial torus bundles over the circle and arise from a quotient-manifold construction applied to suitably chosen discrete isometry groups of diffeomorphically-Euclidean "model" manifolds.