This note aims to demonstrate that every parabolic geometry has a naturally defined per-Courant algebröıd structure. If the geometry is regular, this structure is a Courant algebröıd if and only if the the curvature κ of the Cartan connection vanishes. This note assumes familiarity with both parabolic geometry, and Courant algebröıds. See [ČS00] and [ČG02] for a good introduction to the first c...

A new notion of Cartan pairs as a substitute of notion of vector fields in the noncommutative geometry is proposed. The correspondence between Cartan pairs and differential calculi is established.

Abstract : Let be a 2nd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates. We are going to solve this problem by an applicable method which has been recognized by R. Gardner, and classify them.   

In connection with a seminar on infinitesimal geometry in Princeton, in which I took part, it seemed desirable to clarify the relations between the work of the Princeton school and that of Cartan. With a group © of transformations in m variables £ is associated, in accordance with Klein's Erlanger Program, a homogeneous or plane space 9? of the kind © ; a point of 9? is represented by a set of ...

It is shown that for the generalized rigid body certain Cartan subalgebras (called of coordinate type) of so(n) are equilibrium points for the rigid body dynamics. In the case of so(4) there are three coordinate type Cartan subalgebras which form the set of all the regular equilibria. These coordinate type Cartan subalgebras are the analogues of the three axes of equilibria for the classical ri...

We suggest to use " minimal " choice of quantum gravity theory, that is the quantum field theory, in which space-time is seen as Rie-mannian space and metrics (or vierbein field) is the dynamical variable. We then suggest to use the simplest acceptable action, that is the squared curvature action. The correspondent model is renor-malizable, has the correct classical limit and can be explored us...

We perform the non-linear realisation or the coset formulation of the pure N = 4, D = 5 supergravity. We derive the Lie superalgebra which parameterizes a coset map whose induced Cartan-Maurer form produces the bosonic field equations of the pure N = 4, D = 5 supergravity by canonically satisfying the Cartan-Maurer equation. We also obtain the first-order field equations of the theory as a twis...

If the Einstein-Hilbert action LEH ∝ R is re-expressed in Riemann-Cartan spacetime using the gauge fields of translations, the vierbein field hμ, and the gauge field of local Lorentz transformations, the spin connection Aμα β , there exists a new gauge symmetry which permits reshuffling the torsion, partially or totally, into the Cartan curvature term of the Einstein tensor, and back, via a new...

Wolfgang Kummer was a pioneer of two-dimensional gravity and a strong advocate of the first order formulation in terms of Cartan variables. In the present work we apply Wolfgang Kummer's philosophy, the 'Vienna School approach', to a specific three-dimensional model of gravity, cos-mological topologically massive gravity at the chiral point. Exploiting a new Chern–Simons representation we perfo...