141,136 research outputs found

### The Casimir operator of a metric connection with skew-symmetric torsion

For any triple $(M^n, g, \nabla)$ consisting of a Riemannian manifold and a
metric connection with skew-symmetric torsion we introduce an elliptic, second
order operator $\Omega$ acting on spinor fields. In case of a reductive space
and its canonical connection our construction yields the Casimir operator of
the isometry group. Several non-homogeneous geometries (Sasakian, nearly
K\"ahler, cocalibrated $\mathrm{G}_2$-structures) admit unique connections with
skew-symmetric torsion. We study the corresponding Casimir operator and compare
its kernel with the space of $\nabla$-parallel spinors.Comment: Latex2e, 15 page

### The extended Conformal Einstein field equations with matter: the Einstein-Maxwell field

A discussion is given of the conformal Einstein field equations coupled with
matter whose energy-momentum tensor is trace-free. These resulting equations
are expressed in terms of a generic Weyl connection. The article shows how in
the presence of matter it is possible to construct a conformal gauge which
allows to know \emph{a priori} the location of the conformal boundary. In
vacuum this gauge reduces to the so-called conformal Gaussian gauge. These
ideas are applied to obtain: (i) a new proof of the stability of
Einstein-Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global
stability of purely radiative Einstein-Maxwell spacetimes.Comment: 29 page

### A conformal approach for the analysis of the non-linear stability of pure radiation cosmologies

The conformal Einstein equations for a tracefree (radiation) perfect fluid
are derived in terms of the Levi-Civita connection of a conformally rescaled
metric. These equations are used to provide a non-linear stability result for
de Sitter-like tracefree (radiation) perfect fluid
Friedman-Lema\^{\i}tre-Robertson-Walker cosmological models. The solutions thus
obtained exist globally towards the future and are future geodesically
complete.Comment: 21 page

### Eigenvalue estimates for the Dirac operator depending on the Weyl curvature tensor

We prove new lower bounds for the first eigenvalue of the Dirac operator on
compact manifolds whose Weyl tensor or curvature tensor, respectively, is
divergence free. In the special case of Einstein manifolds, we obtain estimates
depending on the Weyl tensor.Comment: Latex2.09, 9 page

### Does asymptotic simplicity allow for radiation near spatial infinity?

A representation of spatial infinity based in the properties of conformal
geodesics is used to obtain asymptotic expansions of the gravitational field
near the region where null infinity touches spatial infinity. These expansions
show that generic time symmetric initial data with an analytic conformal metric
at spatial infinity will give rise to developments with a certain type of
logarithmic singularities at the points where null infinity and spatial
infinity meet. These logarithmic singularities produce a non-smooth null
infinity. The sources of the logarithmic singularities are traced back down to
the initial data. It is shown that is the parts of the initial data responsible
for the non-regular behaviour of the solutions are not present, then the
initial data is static to a certain order. On the basis of these results it is
conjectured that the only time symmetric data sets with developments having a
smooth null infinity are those which are static in a neighbourhood of infinity.
This conjecture generalises a previous conjecture regarding time symmetric,
conformally flat data. The relation of these conjectures to Penrose's proposal
for the description of the asymptotic gravitational field of isolated bodies is
discussed.Comment: 22 pages, 4 figures. Typos and grammatical mistakes corrected.
Version to appear in Comm. Math. Phy

### Curvature dependent lower bounds for the first eigenvalue of the Dirac operator

Using Weitzenb\"ock techniques on any compact Riemannian spin manifold we
derive inequalities that involve a real parameter and join the eigenvalues of
the Dirac operator with curvature terms. The discussion of these inequalities
yields vanishing theorems for the kernel of the Dirac operator $D$ and lower
bounds for the spectrum of $D^2$ if the curvature satisfies certain conditions.Comment: Latex2e, 14p

### Spin(9)-structures and connections with totally skew-symmetric torsion

We study Spin(9)-structures on 16-dimensional Riemannian manifolds and
characterize the geometric types admitting a connection with totally
skew-symmetric torsion.Comment: Latex2e, 8 page

### Upper bounds for the first eigenvalue of the Dirac operator on surfaces

In this paper we will prove new extrinsic upper bounds for the eigenvalues of
the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow
{\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of
genus zero and genus one. Moreover, we compare the different estimates of the
eigenvalue of the Dirac operator for special families of metrics.Comment: Latex2.09, 23 page

### Almost Hermitian 6-Manifolds Revisited

A Theorem of Kirichenko states that the torsion 3-form of the characteristic
connection of a nearly K\"ahler manifold is parallel. On the other side, any
almost hermitian manifold of type $\mathrm{G}_1$ admits a unique connection
with totally skew symmetric torsion. In dimension six, we generalize
Kirichenko's Theorem and we describe almost hermitian $\mathrm{G}_1$-manifolds
with parallel torsion form. In particular, among them there are only two types
of $\mathcal{W}_3$-manifolds with a non-abelian holonomy group, namely twistor
spaces of 4-dimensional self-dual Einstein manifolds and the invariant
hermitian structure on the Lie group \mathrm{SL}(2, \C). Moreover, we
classify all naturally reductive hermitian $\mathcal{W}_3$-manifolds with small
isotropy group of the characteristic torsion.Comment: 26 pages, revised versio

### The Einstein-Dirac Equation on Riemannian Spin Manifolds

We construct exact solutions of the Einstein-Dirac equation, which couples
the gravitational field with an eigenspinor of the Dirac operator via the
energy-momentum tensor. For this purpose we introduce a new field equation
generalizing the notion of Killing spinors. The solutions of this spinorial
field equation are called weak Killing spinors (WK-spinors). They are special
solutions of the Einstein-Dirac equation and in dimension n=3 the two equations
essentially coincide. It turns out that any Sasakian manifold with Ricci tensor
related in some special way to the metric tensor as well as to the contact
structure admits a WK-spinor. This result is a consequence of the investigation
of special spinorial field equations on Sasakian manifolds (Sasakian
quasi-Killing spinors). Altogether, in odd dimensions a contact geometry
generates a solution of the Einstein-Dirac equation. Moreover, we prove the
existence of solutions of the Einstein-Dirac equations that are not WK-spinors
in all dimensions n > 8.Comment: Latex2.09, 47 page

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