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

We determine the local structure of all pseudo-Riemannian manifolds $(M,g)$ in dimensions $n\ge4$ whose Weyl conformal tensor $W$ 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 $n\ge4$, the metric signature $--...++$, and a sign factor $\epsilon=\pm1$ accounting for semidefiniteness of $W$, then the local-isometry types of our metrics $g$ correspond bijectively to equivalence classes of surfaces $\varSigma$ with equiaffine projectively flat torsionfree connections; the latter equivalence relation is provided by unimodular affine local diffeomorphisms. The surface $\varSigma$ arises, locally, as the leaf space of a codimension-two parallel distribution on $M$, naturally associated with $g$. We exhibit examples in which the leaves of the distribution form a fibration with the total space $M$ and base $\varSigma$, for a closed surface $\varSigma$ 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, rank $W$ = 1, and so they belong to the class mentioned above; on the other hand, the Ricci-recurrent ones have already been classified by the second author.Comment: 39 page

On compact manifolds admitting indefinite metrics with parallel Weyl tensor

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.Comment: 20 page

Compact pseudo-Riemannian manifolds with parallel Weyl tensor

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.Comment: 19 page

International-local Linkages in Multi-stakeholder Partnerships Involved in Reconciliation, Inter-communal Bridge-building and Confidence-building

This paper focuses on the involvement of the international community (international actors) in post-conflict reconstruction in the context of multi-stakeholder partnerships (MSPs) operating in the issue-area of reconciliation, inter-communal bridge-building and confidence-building. In particular, the paper analyses the international-local linkages within the MSPs, and suggests that although the involvement of the international community in post-conflict reconstruction (peace-building) is heavy and indispensable, it is neither straight-forward nor problem-free. In order to understand these linkages in a specific MSP context, a number of factors need to be taken into account and analysed. The paper suggests that at least three levels of analysis are required in order to understand the role of the international community and the international-local linkages in the context of MSPs addressing reconciliation, confidence-building and inter-community bridge-building in a post-conflict context. Firstly, the very complex nature of the international community itself, with many different actors seeking to achieve their own objectives in a very competitive environment; secondly, the very difficult conditions in war-torn societies that are operationally/institutionally unable to begin any peace-building processes on their own; and thirdly, the characteristics (motivations, organisation) of international and domestic actors themselves