3,257 research outputs found

### Deforming a Lie algebra by means of a two form

We consider a vector space V over K=R or C, equipped with a skew symmetric
bracket [.,.]: V x V --> V and a 2-form omega:V x V --> K. A simple change of
the Jacobi identity to the form
[A,[B,C]]+[C,[A,B]]+[B,[C,A]]=omega(B,C)A+omega(A,B)C+omega(C,A)B opens new
possibilities, which shed new light on the Bianchi classification of
3-dimensional Lie algebras.Comment: An error consisting in overlooking few types in the Bianchi
classification is correcte

### Torsion, an alternative to the cosmological constant?

We confront Einstein-Cartan's theory with the Hubble diagram and obtain a
negative answer to the question in the title. Contrary findings in the
literature seem to stem from an error in the field equations.Comment: 10 pages, 1 figure. Version 2 corrects a factor 3 in Cartan's
equations to become

### Differential equations and conformal structures

We provide five examples of conformal geometries which are naturally
associated with ordinary differential equations (ODEs). The first example
describes a one-to-one correspondence between the Wuenschmann class of 3rd
order ODEs considered modulo contact transformations of variables and (local)
3-dimensional conformal Lorentzian geometries. The second example shows that
every point equivalent class of 3rd order ODEs satisfying the Wuenschmann and
the Cartan conditions define a 3-dimensional Lorentzian Einstein-Weyl geometry.
The third example associates to each point equivalence class of 3rd order ODEs
a 6-dimensional conformal geometry of neutral signature. The fourth example
exhibits the one-to-one correspondence between point equivalent classes of 2nd
order ODEs and 4-dimensional conformal Fefferman-like metrics of neutral
signature. The fifth example shows the correspondence between undetermined ODEs
of the Monge type and conformal geometries of signature $(3,2)$. The Cartan
normal conformal connection for these geometries is reducible to the Cartan
connection with values in the Lie algebra of the noncompact form of the
exceptional group $G_2$. All the examples are deeply rooted in Elie Cartan's
works on exterior differential systems.Comment: Some typos in formulae concerning (3,2)-signature conformal metrics
of Section 5.3 were correcte

### Modeling the electron with Cosserat elasticity

We suggest an alternative mathematical model for the electron in dimension
1+2. We think of our (1+2)-dimensional spacetime as an elastic continuum whose
material points can experience no displacements, only rotations. This framework
is a special case of the Cosserat theory of elasticity. Rotations of material
points are described mathematically by attaching to each geometric point an
orthonormal basis which gives a field of orthonormal bases called the coframe.
As the dynamical variables (unknowns) of our theory we choose a coframe and a
density. We then add an extra (third) spatial dimension, extend our coframe and
density into dimension 1+3, choose a conformally invariant Lagrangian
proportional to axial torsion squared, roll up the extra dimension into a
circle so as to incorporate mass and return to our original (1+2)-dimensional
spacetime by separating out the extra coordinate. The main result of our paper
is the theorem stating that our model is equivalent to the Dirac equation in
dimension 1+2. In the process of analyzing our model we also establish an
abstract result, identifying a class of nonlinear second order partial
differential equations which reduce to pairs of linear first order equations

### Estimating the higher symmetric topological complexity of spheres

We study questions of the following type: Can one assign continuously and
$\Sigma_m$-equivariantly to any $m$-tuple of distinct points on the sphere
$S^n$ a multipath in $S^n$ spanning these points? A \emph{multipath} is a
continuous map of the wedge of $m$ segments to the sphere. This question is
connected with the \emph{higher symmetric topological complexity} of spheres,
introduced and studied by I. Basabe, J. Gonz\'alez, Yu. B. Rudyak, and D.
Tamaki. In all cases we can handle, the answer is negative. Our arguments are
in the spirit of the definition of the Hopf invariant of a map $f: S^{2n-1} \to
S^n$ by means of the mapping cone and the cup product.Comment: This version has minor corrections compared to what published in AG

### On a weak Gauss law in general relativity and torsion

We present an explicit example showing that the weak Gauss law of general
relativity (with cosmological constant) fails in Einstein-Cartan's theory. We
take this as an indication that torsion might replace dark matter.Comment: 10 pages. Version 2 corrects a factor 3 in Cartan's equations to
become

### N-dimensional geometries and Einstein equations from systems of PDE's

The aim of the present work is twofold: first, we show how all the
$n$-dimensional Riemannian and Lorentzian metrics can be constructed from a
certain class of systems of second-order PDE's which are in duality to the
Hamilton-Jacobi equation and second we impose the Einstein equations to these
PDE's

### On a certain formulation of the Einstein equations

We define a certain differential system on an open set of $R^6$. The system
locally defines a Lorentzian 4-manifold satisfying the Einstein equations. The
converse statement is indicated and its details are postponed to the
furthcoming paper.Comment: 7 pages, Late

### Paraconformal geometry of $n$th order ODEs, and exotic holonomy in dimension four

We characterise $n$th order ODEs for which the space of solutions $M$ is
equipped with a particular paraconformal structure in the sense of \cite{BE},
that is a splitting of the tangent bundle as a symmetric tensor product of
rank-two vector bundles. This leads to the vanishing of $(n-2)$ quantities
constructed from of the ODE.
If $n=4$ the paraconformal structure is shown to be equivalent to the exotic
${\cal G}_3$ holonomy of Bryant. If $n=4$, or $n\geq 6$ and $M$ admits a
torsion--free connection compatible with the paraconformal structure then the
ODE is trivialisable by point or contact transformations respectively.
If $n=2$ or 3 $M$ admits an affine paraconformal connection with no torsion.
In these cases additional constraints can be imposed on the ODE so that $M$
admits a projective structure if $n=2$, or an Einstein--Weyl structure if
$n=3$. The third order ODE can in this case be reconstructed from the
Einstein--Weyl data.Comment: Theorem 1.2 strengthened and its proof clarified. Theorem 1.3
generalised to all dimensions, updated references, an example of 5th order
ODE on the space of conics in $CP^2$ added, connection with
Doubrov-Wilczynski invariants clarified. Final version, to appear in Journal
of Geometry and Physic

### Axions in gravity with torsion

We study a scenario allowing a solution of the strong charge parity problem
via the Peccei-Quinn mechanism, implemented in gravity with torsion. In this
framework there appears a torsion-related pseudoscalar field known as
Kalb-Ramond axion. We compare it with the so-called Barbero-Immirzi axion
recently proposed in the literature also in the context of the gravity with
torsion. We show that they are equivalent from the viewpoint of the effective
theory. The phenomenology of these torsion-descended axions is completely
determined by the Planck scale without any additional model parameters. These
axions are very light and very weakly interacting with ordinary matter. We
briefly comment on their astrophysical and cosmological implications in view of
the recent BICEP2 and Planck data.Comment: 7 pages, no figures, comments and references added, published versio

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