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

Teichmüller theory and collapse of flat manifolds

We provide an algebraic description of the Teichmüller space and moduli space of flat metrics on a closed manifold or orbifold and study its boundary, which consists of (isometry classes of) flat orbifolds to which the original object may collapse. It is also shown that every closed flat orbifold can be obtained by collapsing closed flat manifolds, and the collapsed limits of closed flat 3-manifolds are classified

Special K\"ahler-Ricci potentials on compact K\"ahler manifolds

A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant $C^\infty$ function $\tau$ such that $J(\nabla\tau)$ is a Killing vector field and, at every point with $d\tau\ne 0$, all nonzero tangent vectors orthogonal to $\nabla\tau$ and $J(\nabla\tau)$ are eigenvectors of both $\nabla d\tau$ and the Ricci tensor. For instance, this is always the case if $\tau$ is a nonconstant $C^\infty$ function on a K\"ahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde g=g/\tau^2$, defined wherever $\tau\ne 0$, is Einstein. (When such $\tau$ exists, $(M,g)$ may be called {\it almost-everywhere conformally Einstein}.) We provide a complete classification of compact K\"ahler manifolds with special K\"ahler-Ricci potentials and use it to prove a structure theorem for compact K\"ahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.Comment: 45 pages, AMSTeX, submitted to Journal f\"ur die reine und angewandte Mathemati

The non-unique Universe

The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical physics, and the implications of Godel's incompleteness theorem for theories of everything. Three conclusions are obtained in the final section: (i) the theory of the structure of our universe might be an undecidable theory, and this constitutes a potential epistemological limit for mathematical physics, but because such a theory must be complete, there is no ontological barrier to the existence of a final theory of everything; (ii) in terms of mathematical logic, there are two different types of multiverse: classes of non-isomorphic but elementarily equivalent models, and classes of model which are both non-isomorphic and elementarily inequivalent; (iii) for a hypothetical theory of everything to have only one possible model, and to thereby negate the possible existence of a multiverse, that theory must be such that it admits only a finite model

Note on (conformally) semi-symmetric spacetimes

We provide a simple proof that conformally semi-symmetric spacetimes are actually semi-symmetric. We also present a complete refined classification of the semi-symmetric spacetimes.Comment: 5 pages, no figure

Weakly Z symmetric manifolds

We introduce a new kind of Riemannian manifold that includes weakly-, pseudo- and pseudo projective- Ricci symmetric manifolds. The manifold is defined through a generalization of the so called Z tensor; it is named "weakly Z symmetric" and denoted by (WZS)_n. If the Z tensor is singular we give conditions for the existence of a proper concircular vector. For non singular Z tensor, we study the closedness property of the associated covectors and give sufficient conditions for the existence of a proper concircular vector in the conformally harmonic case, and the general form of the Ricci tensor. For conformally flat (WZS)_n manifolds, we derive the local form of the metric tensor.Comment: 13 page

On the spectrum of the Page and the Chen-LeBrun-Weber metrics

We give bounds on the first non-zero eigenvalue of the scalar Laplacian for both the Page and the Chen-LeBrun-Weber Einstein metrics. One notable feature is that these bounds are obtained without explicit knowledge of the metrics or numerical approximation to them. Our method also allows the calculation of the invariant part of the spectrum for both metrics. We go on to discuss an application of these bounds to the linear stability of the metrics. We also give numerical evidence to suggest that the bounds for both metrics are extremely close to the actual eigenvalue.Comment: 15 pages, v2 substantially rewritten, section on linear stability added; v3 updated to reflect referee's comments, v4 final version to appear in Ann. Glob. Anal. Geo