2,292 research outputs found
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 biconformal changes of K\"ahler surface metrics
The term "special biconformal change" refers, basically, to the situation
where a given nontrivial real-holomorphic vector field on a complex manifold is
a gradient relative to two K\"ahler metrics, and, simultaneously, an
eigenvector of one of the metrics treated, with the aid of the other, as an
endomorphism of the tangent bundle. A special biconformal change is called
nontrivial if the two metrics are not each other's constant multiples. For
instance, according to a 1995 result of LeBrun, a nontrivial special
biconformal change exists for the conformally-Einstein K\"ahler metric on the
two-point blow-up of the complex projective plane, recently discovered by Chen,
LeBrun and Weber; the real-holomorphic vector field involved is the gradient of
its scalar curvature. The present paper establishes the existence of nontrivial
special biconformal changes for some canonical metrics on Del Pezzo surfaces,
viz. K\"ahler-Einstein metrics (when a nontrivial holomorphic vector field
exists), non-Einstein K\"ahler-Ricci solitons, and K\"ahler metrics admitting
nonconstant Killing potentials with geodesic gradients.Comment: 16 page
Special K\"ahler-Ricci potentials on compact K\"ahler manifolds
A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant
function such that is a Killing vector field
and, at every point with , all nonzero tangent vectors orthogonal
to and are eigenvectors of both and
the Ricci tensor. For instance, this is always the case if is a
nonconstant function on a K\"ahler manifold of complex
dimension and the metric , defined wherever , is Einstein. (When such exists, 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 which are almost-everywhere conformally Einstein.Comment: 45 pages, AMSTeX, submitted to Journal f\"ur die reine und angewandte
Mathemati
Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III
The local structure of the manifolds named in the title is described.
Although curvature homogeneous, they are not, in general, locally homogeneous.
Not all of them are Ricci-flat, which answers an existence question about type
III Jordan-Osserman metrics, raised by Diaz-Ramos, Garcia-Rio and
Vazquez-Lorenzo (2006).Comment: 47 pages; a reference and a grant number were adde
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
Einstein-Weyl structures corresponding to diagonal K\"ahler Bianchi IX metrics
We analyse in a systematic way the four dimensionnal Einstein-Weyl spaces
equipped with a diagonal K\"ahler Bianchi IX metric. In particular, we show
that the subclass of Einstein-Weyl structures with a constant conformal scalar
curvature is the one with a conformally scalar flat - but not necessarily
scalar flat - metric ; we exhibit its 3-parameter distance and Weyl one-form.
This extends previous analysis of Pedersen, Swann and Madsen , limited to the
scalar flat, antiself-dual case. We also check that, in agreement with a
theorem of Derdzinski, the most general conformally Einstein metric in the
family of biaxial K\"ahler Bianchi IX metrics is an extremal metric of Calabi,
conformal to Carter's metric, thanks to Chave and Valent's results.Comment: 15 pages, Latex file, minor modifications, to be published in Class.
Quant. Gra
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