Covariant derivative of the curvature tensor of pseudo-K\"ahlerian manifolds

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-K\"ahlerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-K\"ahlerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group.Comment: the final version accepted to Annals of Global Analysis and Geometr

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

Stability of vector bundles and extremal metrics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46622/1/222_2005_Article_BF01404460.pd

Twistor Bundles, Einstein Equations and Real Structures

We consider sphere bundles P and P' of totally null planes of maximal dimension and opposite self-duality over a 4-dimensional manifold equipped with a Weyl or Riemannian geometry. The fibre product PP' of P and P' is found to be appropriate for the encoding of both the selfdual and the Einstein-Weyl equations for the 4-metric. This encoding is realized in terms of the properties of certain well defined geometrical objects on PP'. The formulation is suitable for both complex- and real-valued metrics. It unifies results for all three possible real signatures. In the purely Riemannian positive definite case it implies the existence of a natural almost hermitian structure on PP' whose integrability conditions correspond to the self-dual Einstein equations of the 4-metric. All Einstein equations for the 4-metric are also encoded in the properties of this almost hermitian structure on PP'.Comment: Paper accepted in Classical and Quantum Gravity, Special issue in honour of Professor Andrzej Trautma