On compact manifolds admitting indefinite metrics with parallel Weyl tensor

TL;DR

This paper explores the topological properties of compact pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric, revealing their fundamental group and bundle structures.

Contribution

It establishes new topological constraints and structural results for such manifolds, especially in the Lorentzian case, extending understanding beyond known examples.

Findings

Vanishing of Euler characteristic and Pontryagin classes

Infinite fundamental group

Lorentzian case manifolds are at least 5-dimensional and admit a circle bundle cover

Abstract

Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classes, and infiniteness of the fundamental group. We also show that, in the Lorentzian case, each of them is at least 5-dimensional and admits a two-fold cover which is a bundle over the circle.