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 nnth 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